When $r = 2m$, Equation (4.16) gives a bijection

$$ \delta_1 \circ \gamma_2 \circ \dots \circ \delta_{2m-1} \circ \gamma_{2m} : \left\{ \begin{matrix} (-v_1, -y) \\ [-u_1, -x) \end{matrix} \right\}^2 \to \left\{ \begin{matrix} (-v_1, -y) \\ [-u_1, -x) \end{matrix} \right\}_{n, 2m} $$

Conjugating by $\nu$ and composing with the map $\zeta$ from Equation (4.12) yields a bijection

$$ \zeta \circ \nu \circ \delta_1 \circ \gamma_2 \circ \dots \circ \delta_{2m-1} \circ \gamma_{2m} \circ \nu : \left\{ \begin{matrix} (x, u_1) \\ [y, v_1) \end{matrix} \right\}^2 \to \left\{ \begin{matrix} (x, u_2) \\ [y, v_2) \end{matrix} \right\}_{n, 2m} $$

that preserves the sum of the entries. Similarly, when $r = 2m + 1$, conjugating the bijection (4.17) with $\nu$ and composing with $\zeta$ produces a bijection

$$ \zeta \circ \nu \circ \delta_1 \circ \gamma_2 \circ \dots \circ \delta_{2m+1} \circ \nu : \left\{ \begin{matrix} (x, u_1) \\ [y, v_1) \end{matrix} \right\}^2 \to \left\{ \begin{matrix} (x, u_2) \\ [y, v_2) \end{matrix} \right\}_{n, 2m+1} $$

In both cases, using Equation (4.7) and Lemma 4.1, we get

$$ H_{A \to B_1, A \to B_2}^{\ge r}(t, q) = \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}}} \left\{ \begin{matrix} (x, u_1) \\ [y, v_1) \end{matrix} \right\}_{n, r} q^{\|\mathbf{c}\|+\|\mathbf{d}\|+\|\mathbf{e}\|+\|\mathbf{f}\|-nz} = f_{r,A,A,B_2,B_1}(t, q), $$

proving Equation (2.18).

**Case 4:** endpoints A and B. The map $\zeta$ from Equation (4.12) restricts to a bijection

$$ \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n, k}^{\ge r \uparrow} \leftarrow \zeta \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n, -k}^{\ge r \downarrow} $$

for any $r \ge 0$ and $k \in \mathbb{Z}$. For $r \ge 1$, we also have

$$ \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_n^{\ge r} = \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_n^{\ge r \uparrow} \sqcup \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_n^{\ge r \downarrow}, $$

and so Equation (4.7) gives

$$ H_{A \to B, A \to B}^{\ge r}(t, q) = 2 \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}}} q^{\|\mathbf{c}\|+\|\mathbf{d}\|+\|\mathbf{e}\|+\|\mathbf{f}\|-nz} \quad (4.18) $$

$$ = 2 \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}}} q^{\|\mathbf{c}\|+\|\mathbf{d}\|+\|\mathbf{e}\|+\|\mathbf{f}\|-nz}. \quad (4.19) $$

Our next goal is to prove that

$$ H_{A \to B, A \to B}^{\ge r}(t, q) + H_{A \to B, A \to B}^{\ge r+1}(t, q) = 2f_{r+1, A, A, B, B}(t, q) \quad (4.20) $$

for all $r \ge 1$.

For arrays with at least $r = 2m$ crossings, Lemmas 4.3 and 4.4 give bijections $\delta_1 \circ \gamma_2 \circ \dots \circ \delta_{2m-1} \circ \gamma_{2m}$:

$$ \begin{align} \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_n^{\ge 2m \uparrow} &\xrightarrow{\gamma_{2m}} \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n-1}^{\ge 2m \uparrow} = \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n-1}^{\ge 2m-1 \downarrow} \nonumber \\ &\xrightarrow{\delta_{2m-1}} \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n, 2}^{\ge 2m-1 \downarrow} = \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n, 2}^{\ge 2m-2 \uparrow} \nonumber \\ &\xrightarrow{\gamma_{2m-2}} \dots \xrightarrow{\delta_1} \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n, 2m}^{\ge 1 \downarrow} = \left\{ \begin{matrix} (x, u) \\ [y, v) \end{matrix} \right\}_{n, 2m}^{\ge 0 \uparrow} \tag{4.21} \end{align} $$
Similarly, for arrays with at least $r = 2m + 1$ crossings, we get bijections $\delta_1 \circ \gamma_2 \circ \cdots \circ \delta_{2m-1} \circ \gamma_{2m} \circ \delta_{2m+1}$:

$$
\begin{align}
\left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_n \stackrel{\ge 2m+1 \downarrow}{\longrightarrow} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,1} &\stackrel{\delta_{2m+1}}{=} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,1} \nonumber \\
&\stackrel{\gamma_{2m}}{\longrightarrow} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,-2} &\stackrel{\ge 2m \uparrow}{=} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,-2} \nonumber \\
&\stackrel{\delta_{2m-1}}{\longrightarrow} \cdots \stackrel{\delta_1}{\longrightarrow} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,2m+1} &\stackrel{\ge 1 \downarrow}{=} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \middle| \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,2m+1}. \tag{4.22}
\end{align}
$$

In both cases, we can compose these bijections with $\zeta \circ \gamma_0$:

$$
\left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,r} \stackrel{\ge 0 \uparrow}{\longrightarrow} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,-r-1} \stackrel{\zeta}{\longrightarrow} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,r+1} \stackrel{\ge 0 \downarrow}{=} \quad (4.23)
$$

Composing (4.21) with (4.23), where $r = 2m$, and using Equation (4.18), we get

$$
H_{\tilde{A} \to B, A \to B}^{\ge 2m}(t, q) = 2 \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}} \mid \mathbf{e} \\ \mathbf{f} \in \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,2m+1}} q^{\|\mathbf{c}\| + \|\mathbf{d}\| + \|\mathbf{e}\| + \|\mathbf{f}\| - nz}
$$

for $m \ge 1$. Similarly, the bijection (4.22) and Equation (4.19), where $r = 2m + 1$, give

$$
H_{\tilde{A} \to B, A \to B}^{\ge 2m+1}(t, q) = 2 \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}} | \mathbf{e} \\ \mathbf{f} \in \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,2m+1}} q^{\|\mathbf{c}\| + \|\mathbf{d}\| + \|\mathbf{e}\| + \|\mathbf{f}\| - nz}.
$$

Adding the last two equations, using the fact that

$$
\left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,k} \stackrel{\ge 0 \uparrow}{\sqcup} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,k} \stackrel{\ge 0 \downarrow}{=} \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,k}
$$

for all $k \ne 0$, and applying Lemma 4.1, we obtain a proof of Equation (4.20) for $r = 2m$.

On the other hand, composing (4.22) (with $m-1$ playing the role of $m$) with (4.23)
(with $r = 2m - 1$) and using Equation (4.19), we get

$$
H_{\tilde{A} \to B, A \to B}^{\ge 2m-1}(t, q) = 2 \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}} | \mathbf{e} \\ \mathbf{f} \in \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,2m}} q^{\|\mathbf{c}\| + \|\mathbf{d}\| + \|\mathbf{e}\| + \|\mathbf{f}\| - nz}
$$

for $m \ge 1$. Similarly, the bijection (4.21) and Equation (4.18), where $r = 2m$, give

$$
H_{\tilde{A} \to B, A \to B}^{\ge 2m}(t, q) = 2 \sum_{n \ge 0} t^n \sum_{\substack{\mathbf{c} \\ \mathbf{d}} | \mathbf{e} \\ \mathbf{f} \in \left\{ \begin{array}{c} (x,u) \\ [y,v] \end{array} \right\}_{n,2m}} q^{\|\mathbf{c}\| + \|\mathbf{d}\| + \|\mathbf{e}\| + \|\mathbf{f}\| - nz}.
$$

Adding the last two equations and applying Lemma 4.1, we obtain a proof of Equation (4.20)
for $r = 2m - 1$.

Solving Equation (4.20) for $H_{\tilde{A} \to B, A \to B}^{\ge r}(t, q)$ and iterating, we obtain

$$
H_{\tilde{A} \to B, A \to B}^{\ge r}(t, q) = 2(f_{r+1, A, A, B, B}(t, q) - f_{r+2, A, A, B, B}(t, q) + f_{r+3, A, A, B, B}(t, q) - \dots)
$$

which proves Equation (2.20) for $r \ge 1$. The case $r = 0$ follows immediately from
Equation (4.7) and Lemma 4.1.

**Acknowledgments.** The author is grateful to Christian Krattenthaler for posing the question of whether the results from [3] could be refined by the number of descents.
REFERENCES

[1] George E. Andrews. *The Theory of Partitions*. Addison-Wesley, 1976.

[2] Sylvie Corteel, Sergi Elizalde, and Carla D. Savage. Generalized rank parity blocks in partitions. In preparation.

[3] Sergi Elizalde. Counting lattice paths by crossings and major index I: the corner-flipping bijections. *Comb. Theory*, 2(2):Article #14, 37 pp., 2022.

[4] Ora Engelberg. On some problems concerning a restricted random walk. *J. Appl. Probability*, 2:396-404, 1965.

[5] William Feller. The numbers of zeros and of changes of sign in a symmetric random walk. *Enseign. Math.* (2), 3:229-235, 1957.

[6] William Feller. *An Introduction to Probability Theory and Its Applications. Vol. I*. John Wiley & Sons, third edition, 1968.

[7] Michael E. Fisher. Walks, walls, wetting, and melting. *J. Statist. Phys.*, 34(5-6):667-729, 1984.

[8] Johannes Fürlinger and Joseph Hofbauer. *q-Catalan numbers*. *J. Combin. Theory Ser. A*, 40(2):248-264, 1985.

[9] Ira Gessel and Gérard Viennot. Binomial determinants, paths, and hook length formulae. *Adv. in Math.*, 58(3):300-321, 1985.

[10] Malcolm Kern and Stanley Walter. Ballot theorem and lattice path crossings. *Canad. J. Statist.*, 6(1):87-90, 1978.

[11] Christian Krattenthaler. Counting nonintersecting lattice paths with turns. *Sém. Lothar. Combin.*, 34:Article B34i, 17 pp., 1995.

[12] Christian Krattenthaler. *The Major Counting of Nonintersecting Lattice Paths and Generating Functions for Tableaux*. *Mem. Amer. Math. Soc.*, 115, 1995.

[13] Christian Krattenthaler. The enumeration of lattice paths with respect to their number of turns. In *Advances in Combinatorial Methods and Applications to Probability and Statistics*, Stat. Ind. Technol., pages 29-58. Birkhäuser, 1997.

[14] Christian Krattenthaler. Lattice path enumeration. In *Handbook of Enumerative Combinatorics*, Discrete Math. Appl., pages 589-678. CRC Press, 2015.

[15] Christian Krattenthaler and Sri Gopal Mohanty. On lattice path counting by major index and descents. *European J. Combin.*, 14(1):43-51, 1993.

[16] Devadatta M. Kulkarni. Counting of paths and coefficients of the Hilbert polynomial of a determinantal ideal. *Discrete Math.*, 154(1-3):141-151, 1996.

[17] Bernt Lindström. On the vector representations of induced matroids. *Bull. London Math. Soc.*, 5:85-90, 1973.

[18] Percy A. MacMahon. *Combinatory Analysis*. Cambridge Univ. Press, 1915-1916. Reprinted by Chelsea in 1960.

[19] Sri Gopal Mohanty. *Lattice Path Counting and Applications*. Academic Press, 1979.

[20] Bruce E. Sagan and Carla D. Savage. Mahonian pairs. *J. Combin. Theory Ser. A*, 119(3):526-545, 2012.

[21] Kanwar Sen. On some combinatorial relations concerning the symmetric random walk. *Magyar Tud. Akad. Mat. Kutató Int. Közl.*, 9:335-357, 1965.

[22] Seunghyun Seo and Ae Ja Yee. Enumeration of partitions with prescribed successive rank parity blocks. *J. Combin. Theory Ser. A*, 158:12-35, 2018.

[23] Michael Z. Spivey. Enumerating lattice paths touching or crossing the diagonal at a given number of lattice points. *Electron. J. Combin.*, 19(3):Article #24, 6 pp., 2012.
# THREE FAMILIES OF q-LOMMEL POLYNOMIALS

JANG SOO KIM¹ ID AND DENNIS STANTON² ID

¹DEPARTMENT OF MATHEMATICS, SUNGKYUNKWAN UNIVERSITY (SKKU), SUWON, GYEONGGI-DO 16419, SOUTH KOREA; https://jangsookim.github.io/

²SCHOOL OF MATHEMATICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINNESOTA 55455, USA; https://www-users.cse.umn.edu/~stant001/

**Abstract.** Three $q$-versions of Lommel polynomials are studied. Included are explicit representations, recurrences, continued fractions, and connections to associated Askey-Wilson polynomials. Combinatorial results are emphasized, including a general theorem when $R_I$ moments coincide with orthogonal polynomial moments. The combinatorial results use weighted Motzkin paths, Schröder paths, and parallelogram polyominoes.

**Keywords:** Lommel polynomial, Bessel function, orthogonal polynomial.

## 1. INTRODUCTION

Lehmer [27] used the following Bessel function identity to study zeros of Bessel functions

$$ \frac{J_{\nu+1}(x)}{J_{\nu}(x)} = 2 \sum_{n=1}^{\infty} \sigma_{2n}(\nu) x^{2n-1}, \qquad (1.1) $$

where $\sigma_{2n}(\nu)$ is the $2n^{th}$ power sum of the inverses of the positive zeros $j_{\nu,k}$ of $J_{\nu}(x)$:

$$ \sigma_{2n}(\nu) = \sum_{k=1}^{\infty} j_{\nu,k}^{-2n}. \qquad (1.2) $$

Lehmer noted that $\sigma_{2n}(\nu)$ is a rational function of $\nu$, with a predictable denominator, and a numerator with nonnegative coefficients. Kishore [19] proved Lehmer's positivity conjecture. Lalanne ([25, Proposition 3.6], [26, Theorem 4.7]) proved $q$-versions of Kishore's result using weighted binary trees and also weighted Dyck paths.

The above series is related to the Lommel polynomials $L_{n,\nu}$, which are orthogonal polynomials with respect to the linear functional

$$ \mathcal{L}(P(x)) := 2(\nu + 1) \sum_{k=1}^{\infty} \left( P\left(j_{\nu,k}^{-1}\right) + P\left(-j_{\nu,k}^{-1}\right) \right) j_{\nu,k}^{-2}, $$

i.e., $\mathcal{L}(L_{n,\nu}(x)L_{m,\nu}(x)) = 0$ if $n \neq m$ and $\mathcal{L}(1) = 1$; see [15, Eq. (6.5.17)]. Thus $\sigma_{2n}(\nu)$ in (1.2) is the $(2n-2)^{th}$ moment for the Lommel polynomials, while (1.1) is the Lommel moment generating function.

The purpose of this paper is to study three sets of $q$-Lommel polynomials, whose moment generating functions are quotients of $q$-Bessel functions. These polynomials were analytically studied by Ismail [15], Koelink and Van Assche [23], and Koelink [21]. In this paper we concentrate on the combinatorial aspect of these three sets of $q$-Lommel polynomials.
The literature already contains some combinatorial results on the quotient of Bessel functions and the quotient of $q$-Bessel functions. Delest and Fédou [9] showed that a generating function for parallelogram polyominoes can be written as a ratio of Jackson's third $q$-Bessel functions. Bousquet-Mélou and Viennot [4] generalized their result by adding one more parameter. A recounting of the history of the combinatorics of the $q$-analogue of the quotient of Bessel functions may be found in [3, Section 1] (see also [26, Section 4]). It includes results by Klarner and Rivest [20, Eq. (19)], Fédou [12], Lalanne [25, 26], Brak and Guttmann [5], and Barcucci et al. [1, Corollary 3.5], [2, Theorems 4.3 and 5.3].

In this paper we put these results in perspective by relating them to $q$-Lommel polynomials. The moment generating function has a continued fraction expansion. Using the general theory of orthogonal and type $R_I$ polynomials we give finite versions of the infinite continued fractions. We show that a generating function for bounded diagonal parallelogram polyominoes is given by a ratio of $q$-Lommel polynomials, which is a finite version of the result of Bousquet-Mélou and Viennot [4].

Even though the Lommel polynomials have a hypergeometric representation as a $_2F_3$, they do not appear in the Askey scheme. In this paper we rectify this, by realizing two sets of $q$-Lommel polynomials as limiting cases of associated Askey-Wilson polynomials. One may ask for an associated Askey scheme which contains this limiting case (see Problem 8.7).

The paper is organized in the following way. In Section 2 we define the three sets of $q$-Lommel polynomials using three-term recurrence relations. The classical connection between these polynomials and $q$-Bessel functions is given in Section 3. The associated Askey-Wilson polynomials are reviewed in Section 4, along with explicit limiting cases to the $q$-Lommel polynomials; see Theorems 4.7 and 4.8. In Section 5 we independently prove the continued fraction expansions for the moment generating functions, and give two surprising equalities of continued fractions in Corollary 5.6 and Theorem 5.12. Combinatorial interpretations of these continued fractions are given in Section 6; see Theorem 6.9 and Corollary 6.11. A general combinatorial result for the concurrence of type $R_I$ moments and orthogonal polynomials moments is given in Section 7; see Theorem 7.2. In Section 8 we propose some open problems.

We use the standard notations $_pF_q$ for hypergeometric series and $_p\phi_q$ for basic hypergeometric series (also sometimes called $q$-hypergeometric series) [14].

## 2. $q$-LOMMEL POLYNOMIALS

In this section we give the defining recurrence relations for the Lommel, the classical $q$-Lommel, the even-odd $q$-Lommel, and the type $R_I$ $q$-Lommel polynomials.

**Definition 2.1.** The monic Lommel polynomials $h_n(x; c)$ are defined by

$$h_{n+1}(x; c) = xh_n(x; c) - \frac{1}{(c+n)(c+n-1)}h_{n-1}(x; c), n \ge 0, \quad h_{-1}(x; c) = 0, h_0(x; c) = 1.$$

We consider three versions of $q$-Lommel polynomials.

**Definition 2.2** ([15, § 14.4]). The classical $q$-Lommel polynomials are defined by

$$h_{n+1}(x; c, q) = xh_n(x; c, q) - \lambda_n h_{n-1}(x; c, q), \quad n \ge 0, \quad h_{-1}(x; c, q) = 0, \quad h_0(x; c, q) = 1,$$ 
where $\lambda_n = \frac{cq^{n-1}}{(1 - cq^{n-1})(1 - cq^n)}$.
**Definition 2.3.** The even-odd q-Lommel polynomials are defined by

$$p_{n+1}(x; c, q) = xp_n(x; c, q) - \lambda_n p_{n-1}(x; c, q), \quad n \ge 0, \quad p_{-1}(x; c, q) = 0, \quad p_0(x; c, q) = 1,$$

$$\text{where } \lambda_{2n} = \frac{cq^{3n-1}}{(1 - cq^{2n-1})(1 - cq^{2n})}, \qquad \lambda_{2n+1} = \frac{q^n}{(1 - cq^{2n})(1 - cq^{2n+1})}.$$

Note that each polynomial $h_n$ and $p_n$ may be considered as a q-analogue of the classical Lommel polynomials since

$$\lim_{q \to 1} (1-q)^n h_n(x/(1-q); q^c, q) = h_n(x; c), \quad \lim_{q \to 1} (1-q)^n p_n(x/(1-q); q^c, q) = h_n(x; c).$$

**Definition 2.4.** The type $R_I$ q-Lommel polynomials are defined by

$$r_{n+1}(x; c, q) = (x - b_n)r_n(x; c, q) - xa_n r_{n-1}(x; c, q), \quad r_{-1}(x; c, q) = 0, \quad r_0(x; c, q) = 1, \qquad (2.1)$$

$$\text{where } b_n = \frac{q^n}{1 - cq^n}, \qquad a_n = \frac{cq^{2n-1}}{(1 - cq^{n-1})(1 - cq^n)}.$$

Note that if

$$\hat{r}_n(x; c) = \lim_{q \to 1} (1-q)^{2n} r_n(x/(1-q)^2; q^c, q),$$

then

$$\hat{r}_{n+1}(x; c) = x\hat{r}_n(x; c) - \frac{x}{(c+n-1)(c+n)}\hat{r}_{n-1}(x; c). \qquad (2.2)$$

The polynomials $\hat{r}_n(x; c)$ in (2.2) are closely related to the monic Lommel polynomials. For example, it is known that their moments are the same; see (7.2).

Koelink and Van Assche study the even-odd and the type $R_I$ q-Lommel polynomials<fnref n="1" /> in [23, Section 4], and Koelink continues this analytic study in [21].

Orthogonality relations for the classical q-Lommel are in [15, Theorem 14.4.3], while those for the even-odd q-Lommel and the type $R_I$ q-Lommel are in [23, Theorem 4.2] and [23, Theorem 3.4].

### 3. q-BESSEL FUNCTIONS AND q-LOMMEL POLYNOMIALS

In this section we give the recurrence relation which connects q-Bessel functions to the classical q-Lommel polynomials and the type $R_I$ q-Lommel polynomials.

**Definition 3.1.** The Bessel functions $J_\nu(x)$ are defined by

$$J_\nu(z) = \frac{(z/2)^\nu}{\Gamma(\nu + 1)} \sum_{n \ge 0} \frac{(-z^2/4)^n}{n!(\nu + 1)_n}.$$

**Definition 3.2** ([6, p. 188, (6.2)]). The classical Lommel polynomials $L_{n,\nu}(z)$ are (non-monic) polynomials in $z^{-1}$ defined by $L_{0,\nu}(z) = 1$, $L_{1,\nu}(z) = 2\nu/z$, and

$$L_{n+1,\nu}(z) = \frac{2(n+\nu)}{z} L_{n,\nu}(z) - L_{n-1,\nu}(z), \quad n \ge 1.$$

Equivalently,

$$h_n(x; c) = L_{n,c}(2/x)/(c)_n.$$

<footnote n="1">¹Continued fractions of type R and the corresponding orthogonal polynomials of type $R_I$ or $R_{II}$ were introduced by Ismail and Masson in [16], the notation being a mnemonic for rational interpolation.</footnote>
The Bessel function satisfy $J_{\nu+1}(z) = \frac{2\nu}{z}J_{\nu}(z) - J_{\nu-1}(z)$. Iterating this recurrence offers the following connection with Lommel polynomials.

**Proposition 3.3** ([6, p. 187]). *The Bessel functions and the classical Lommel polynomials are related by the recurrence*

$$J_{\nu+n}(z) = L_{n,\nu}(z)J_{\nu}(z) - L_{n-1,\nu+1}(z)J_{\nu-1}(z).$$

**Definition 3.4.** Jackson's first q-Bessel function $J_{\nu}^{(1)}(z; q)$ and second q-Bessel function $J_{\nu}^{(2)}(z; q)$ are defined by

$$J_{\nu}^{(1)}(z; q) = \frac{(q^{\nu+1}; q)_{\infty}}{(q; q)_{\infty}} (z/2)^{\nu} {}_2\phi_1 (0, 0; q^{\nu+1}; q, -z^2/4), \\ J_{\nu}^{(2)}(z; q) = \frac{(q^{\nu+1}; q)_{\infty}}{(q; q)_{\infty}} (z/2)^{\nu} {}_0\phi_1 (-; q^{\nu+1}; q, -q^{\nu+1}z^2/4).$$

In this paper we consider only the first and third q-Bessel function, as the second q-Bessel can be obtained from the first by changing q to $q^{-1}$. Recall that we consider formal power series in z, and have no restriction on q.

**Proposition 3.5** ([15, Eq. (14.4.1)]). *The first q-Bessel functions satisfy*

$$q^{n\nu+\binom{n}{2}} J_{\nu+n}^{(1)}(x;q) = L_{n,\nu}^{(1)}(x;q)J_{\nu}^{(1)}(x;q) - L_{n-1,\nu+1}^{(1)}(x;q)J_{\nu-1}^{(1)}(x;q). \quad (3.1)$$

where $L_{0,\nu}^{(1)}(x;q) = 1$, $L_{1,\nu}^{(1)}(x;q) = 2(1-q^{n+\nu})/x$, and

$$\frac{2}{x}(1 - q^{n+\nu})L_{n,\nu}^{(1)}(x;q) = L_{n+1,\nu}^{(1)}(x;q) + q^{n+\nu-1}L_{n-1,\nu}^{(1)}(x;q), \quad n \ge 1.$$

Again, we need a rescaling to obtain the classical q-Lommel polynomials,

$$h_n(x; c, q) = L_{n,\nu}^{(1)}(2/x; q)/(q^{\nu}; q)_n, \quad c = q^{\nu}.$$

**Definition 3.6.** The Jackson's third q-Bessel functions $J_{\nu}^{(3)}(z; q)$ are defined by

$$J_{\nu}^{(3)}(z; q) = \frac{(q^{\nu+1}; q)_{\infty} z^{\nu}}{(q; q)_{\infty}} {}_1\phi_1 (0; q^{\nu+1}; q, qz^2).$$

Define the Laurent polynomials $L_{m,\nu}^{(3)}(z; q)$ by

$$L_{m+1,\nu}^{(3)}(z;q) = (z + z^{-1}(1 - q^{\nu+m})) L_{m,\nu}^{(3)}(z;q) - L_{m-1,\nu}^{(3)}(z;q).$$

We rescale these Laurent polynomials to obtain polynomials

$$r_n^{(3)}(x; c, q) := \frac{x^{n/2}}{(q^{-\nu}; q^{-1})_n} L_{n,\nu}^{(3)}(x^{-1/2}; q^{-1}), \quad c = q^{\nu}. \quad (3.2)$$

Then $r_n^{(3)}(x; c, q)$ are the type $R_I$ polynomials defined by $r_{-1}^{(3)}(x; c, q) = 0$, $r_0^{(3)}(x; c, q) = 1$, and

$$r_{n+1}^{(3)}(x; c, q) = (x - \hat{b}_n) r_n^{(3)}(x; c, q) - x \hat{a}_n r_{n-1}^{(3)}(x; c, q), \quad n \ge 0, \\ \text{where } \hat{b}_n = \frac{c q^n}{1 - c q^n}, \quad \hat{a}_n = \frac{c^2 q^{2n-1}}{(1 - c q^{n-1})(1 - c q^n)}.$$

Using the recurrences one can easily check that

$$r_n(x; c, q) = \frac{r_n^{(3)}(cx; c, q)}{c^n},$$
where $r_n(x; c, q)$ are the type $R_I$ q-Lommel polynomials $r_n(x; c, q)$ in Definition 2.4.

Koelink and Swarttouw [22, Eq. (4.12)] showed that the third q-Bessel functions satisfy the following property analogous to (3.3) and (3.1).

**Proposition 3.7.** *The third q-Bessel functions satisfy*

$$J_{\nu+m}^{(3)}(z; q) = L_{m,\nu}^{(3)}(z; q)J_{\nu}^{(3)}(z; q) - L_{m-1,\nu+1}^{(3)}(z; q)J_{\nu-1}^{(3)}(z; q).$$

Koelink and Swarttouw [22, Eq. (4.24)] also showed that

$$\lim_{m \to \infty} z^m R_{m, \nu}^{(3)}(z; q) = \frac{(q; q)_\infty z^{1-\nu}}{(z^2; q)_\infty} J_{\nu-1}^{(3)}(z; q),$$

which implies

$$\lim_{m \to \infty} \frac{R_{m, \nu+2}^{(3)}(z; q)}{R_{m+1, \nu+1}^{(3)}(z; q)} = \frac{J_{\nu+1}^{(3)}(z; q)}{J_{\nu}^{(3)}(z; q)}. \qquad (3.3)$$

By (3.2) and (3.3) we have

$$\frac{J_{\nu+1}^{(3)}(x^{1/2}; q^{-1})}{J_{\nu}^{(3)}(x^{1/2}; q^{-1})} = \lim_{n \to \infty} \frac{-q^{\nu+1} r_n^{(3)}(x^{-1}; q^{\nu+2}, q)}{x^{1/2}(1-q^{\nu+1}) r_{n+1}^{(3)}(x^{-1}; q^{\nu+1}, q)}. \qquad (3.4)$$

The q-Bessel function relation for the even-odd q-Lommel polynomials which corresponds to Proposition 3.5 is given in [23, Proposition 4.1].

## 4. q-LOMMEL POLYNOMIALS AND THE ASKEY SCHEME

The q-Lommel polynomials do not appear in the Askey scheme. In this section we realize both the classical q-Lommel and the even-odd q-Lommel polynomials as limiting cases of the associated Askey-Wilson polynomials; see Theorems 4.7 and 4.8. We then use results of Ismail and Masson [16] to give explicit formulas for each polynomial. Finally, we prove that the moments for even-odd q-Lommel and the type $R_I$ q-Lommel agree; see Theorem 4.14.

An explicit formula for the Lommel polynomial $h_n(x; c)$ is

$$h_n(x; c) = x^n {}_2F_3(-n/2, (1-n)/2; c, 1-c-n, -n; -4/x^2).$$

In this section we give explicit formulas for our three families of q-Lommel polynomials. The classical q-Lommel polynomials have a corresponding single sum formula [15, Theorem 14.4.1]:

$$h_n(x; c, q) = \frac{1}{(c; q)_n} \sum_{j=0}^{\lfloor n/2 \rfloor} \frac{(-1)^j (c, q; q)_{n-j}}{(q, c; q)_j (q; q)_{n-2j}} x^{n-2j} c^j q^{j(j-1)}.$$

Here are the main results for the even-odd q-Lommel polynomials.

**Theorem 4.1.** *The even-odd q-Lommel polynomials have the explicit formula*

$$p_{2n}(x; c, q) = (-1)^n \frac{q^{\binom{n}{2}}}{(c; q)_{2n}} \sum_{k=0}^{n} \frac{(q^{-n}, cq^n, c; q)_k}{(q; q)_k} q^k x^{2k} \\ \times \sum_{s=0}^{n-k} \frac{(cq^{k-1}; q)_s}{(q; q)_s} \frac{1 - cq^{k-1+2s}}{1 - cq^{k-1}} \frac{(cq^{n+k}, q^{k-n}, q^k; q)_s}{(q^{-n}, cq^n, c; q)_s} c^s q^{-sk+s(s-1)}.$$
**Theorem 4.2.** The odd even-odd q-Lommel polynomials have the explicit formula

$$p_{2n+1}(x; c, q) = (-c)^n \frac{q^{n^2+\binom{n+1}{2}}}{(cq; q)_{2n}} \sum_{k=0}^{n} \frac{(q^{-n}, cq^{n+1}, cq; q)_k}{(q; q)_k} c^{-k} q^{-k^2} x^{2k+1} \\ \times \sum_{s=0}^{n-k} \frac{(cq^k; q)_s}{(q; q)_s} \frac{1 - cq^{k+2s}}{1 - cq^k} \frac{(cq^{n+k+1}, q^{k-n}, q^{k+1}; q)_s}{(q^{-n}, cq^{n+1}, c; q)_s} c^s q^{-(3k+2)s-s(s-1)}.$$

Note that the inner sums in Theorems 4.1 and 4.2 are basic hypergeometric series. The coefficient of $x^{2k}$ in Theorem 4.1 and $x^{2k+1}$ in Theorem 4.2 are terminating basic hypergeometric series.

In order to prove Theorems 4.1 and 4.2, we first write the even even-odd polynomials as orthogonal polynomials in $x^2$ using the odd-even trick. Then we realize the new polynomials as limiting cases of associated Askey-Wilson polynomials, for which explicit formulas are known. The same method will work for the odd even-odd polynomials.

We begin with the associated Askey-Wilson polynomials. First, recall that the monic Askey-Wilson polynomials satisfy the following three-term recurrence (which, as claimed by Favard's theorem, is characterizing orthogonal polynomials):

$$p_{n+1}(x) = (x - b_n)p_n(x) - \lambda_n p_{n-1}(x), \quad n \ge 1, \qquad (4.1)$$

where

$$b_n = \frac{1}{2}(a + a^{-1} - A_n - C_n), \quad \lambda_n = \frac{1}{4}A_{n-1}C_n, \\ A_n = \frac{(1 - abq^n)(1 - acq^n)(1 - adq^n)(1 - abcdq^{n-1})}{a(1 - abcdq^{2n-1})(1 - abcdq^{2n})}, \\ C_n = \frac{a(1 - q^n)(1 - bcq^{n-1})(1 - bdq^{n-1})(1 - cdq^{n-1})}{(1 - abcdq^{2n-2})(1 - abcdq^{2n-1})}.$$

Then, the associated Askey-Wilson polynomials are defined by the same three-term recurrence relation (4.1), with $q^n$ replaced by $\alpha q^n$.

**Definition 4.3.** The associated Askey-Wilson polynomials $p_n^{(\alpha)}(x)$ are defined as a solution to

$$p_{n+1}^{(\alpha)}(x) = (x - b_n(\alpha))p_n^{(\alpha)}(x) - \lambda_n(\alpha)p_{n-1}^{(\alpha)}(x), \quad n \ge 1, \qquad (4.2)$$

$$b_n(\alpha) = \frac{1}{2}(a + a^{-1} - A_n(\alpha) - C_n(\alpha)), \quad \lambda_n(\alpha) = \frac{1}{4}A_{n-1}(\alpha)C_n(\alpha), \\ A_n(\alpha, q) = \frac{(1 - ab\alpha q^n)(1 - ac\alpha q^n)(1 - ad\alpha q^n)(1 - abcd\alpha q^{n-1})}{a(1 - abcd\alpha^2 q^{2n-1})(1 - abcd\alpha^2 q^{2n})}, \\ C_n(\alpha, q) = \frac{a(1 - \alpha q^n)(1 - bc\alpha q^{n-1})(1 - bd\alpha q^{n-1})(1 - cd\alpha q^{n-1})}{(1 - abcd\alpha^2 q^{2n-2})(1 - abcd\alpha^2 q^{2n-1})}.$$

There are two linearly independent solutions to (4.2), depending on the initial conditions. Ismail and Rahman [15, Eq. (4.15), Eq. (8.9)] gave these two independent solutions as double sums, the inner sum being a very-well-poised $_{10}W_9$.
**Theorem 4.4.** Two linearly independent solutions $\psi_n^{(\alpha, \epsilon)}(x, q)$, $\epsilon = 1, 2$ to (4.2) are given by

$$ \psi_n^{(\alpha, \epsilon)}(x; q) = K_n \sum_{k=0}^{n} \frac{(q^{-n},abcd\alpha^2q^{n-1},abcd\alpha^2/q,az,a/z;q)_k}{(q,ab\alpha,ac\alpha,ad\alpha,abcd\alpha/q;q)_k} q^k \\ \times {}_{10}W_9 \left(abcd\alpha^2q^{k-2};\alpha,bc\alpha/q,bd\alpha/q,cd\alpha/q,S,abcd\alpha^2q^{n+k-1},q^{k-n};q;T \right) $$

where

$$ K_n = (2a)^{-n} \frac{(ab\alpha, ac\alpha, ad\alpha,abcd\alpha/q;q)_n}{(abcd\alpha^2q^{n-1},abcd\alpha^2/q;q)_n} $$

and the two choices for $\epsilon$ correspond to

$$ (S,T) = (q^{k+1}, a^2), \text{ for } \epsilon = 1, (S,T) = (q^k, qa^2), \text{ for } \epsilon = 2. $$

We next explain how Theorem 4.1 follows from Theorem 4.4. First we rewrite the recurrence relation [6] in terms of polynomials in $x^2$. Let $t_n(x)$ and $s_n(x)$ be the polynomials satisfying

$$ \begin{aligned} p_{2n}(x; c, q) &= t_n(x^2), \\ p_{2n+1}(x; c, q) &= xs_n(x^2). \end{aligned} $$

**Proposition 4.5.** We have

$$ t_{n+1}(x) = (x - B_n)t_n(x) - \Lambda_n t_{n-1}(x), \quad t_{-1} = 0, \quad t_0(x) = 1. $$

where

$$ \begin{aligned} B_0 &= \frac{1}{(1-c)(1-cq)}, \\ B_n &= \lambda_{2n} + \lambda_{2n+1}, \quad n \ge 1, \\ \Lambda_n &= \lambda_{2n-1}\lambda_{2n}, \quad n \ge 1. \end{aligned} $$

**Proposition 4.6.** We have

$$ s_{n+1}(x) = (x - B_n)s_n(x) - \Lambda_n s_{n-1}(x), \quad s_{-1} = 0, \quad s_0(x) = 1. $$

where

$$ \begin{aligned} B_n &= \lambda_{2n+2} + \lambda_{2n+1}, \quad n \ge 1, \\ \Lambda_n &= \lambda_{2n+1}\lambda_{2n}, \quad n \ge 1. \end{aligned} $$

We shall obtain the recurrence relations in Propositions 4.5 and 4.6 by an appropriate limiting case of Theorem 4.4. Our goal is to obtain $(A_n, C_n) = (\lambda_{2n+1}, \lambda_{2n})$ for $t_n(x)$ and $(A_n, C_n) = (\lambda_{2n+2}, \lambda_{2n+1})$ for $s_n(x)$. Then we match the initial conditions to find the correct linear combination of the two solutions.

First choosing $a = c^{-1}q^{-1}\alpha$, $b = c = d = 1/\alpha$, we obtain

$$ \begin{aligned} A_n(\alpha, 1/q) &= \frac{\alpha(1 - cq^{n+1}/\alpha)^3(1 - \alpha cq^n)}{cq(1 - cq^{2n})(1 - cq^{2n+1})}, \\ C_n(\alpha, 1/q) &= \frac{cq(1 - q^n/\alpha)(1 - \alpha q^{n-1})^3}{\alpha(1 - cq^{2n-1})(1 - cq^{2n})}. \end{aligned} $$
By rescaling $x$ by $B\alpha^2x/2$, i.e., $\hat{p}_n(x) = 2^n\alpha^{-2n}B^{-n}p_n^{(\alpha)}(B\alpha^2x/2)$, we have

$$ \hat{p}_{n+1}(x) = (x - \hat{b}_n(\alpha))\hat{p}_n(x) - \hat{\lambda}_n(\alpha)\hat{p}_{n-1}(x), $$

$$ \hat{b}_n(\alpha) = \frac{1}{B\alpha^2} \left( \frac{cq}{\alpha} + \frac{\alpha}{cq} - A_n(\alpha, 1/q) - C_n(\alpha, 1/q) \right), $$

$$ \hat{\lambda}_n(\alpha) = \frac{1}{B^2\alpha^4} A_n(\alpha, 1/q)C_n(\alpha, 1/q). $$

If $\alpha \to \infty$, the first two terms in $\hat{b}_n(\alpha)$ vanish. Choosing $B = 1/q$, we obtain the desired values for Proposition 4.5

$$ \lim_{\alpha \to \infty} \frac{-1}{B\alpha^2} A_n(\alpha, 1/q) = \frac{q^n}{(1 - cq^{2n})(1 - cq^{2n+1})} = \lambda_{2n+1}, $$

$$ \lim_{\alpha \to \infty} \frac{-1}{B\alpha^2} C_n(\alpha, 1/q) = \frac{cq^{3n-1}}{(1 - cq^{2n-1})(1 - cq^{2n})} = \lambda_{2n}. $$

The first degree limiting polynomial matches the second Ismail–Rahman solution in Theorem 4.4 with $(a, b, c, d) = (\alpha/cq, 1/\alpha, 1/\alpha, 1/\alpha)$,

$$ x - \frac{1}{(1 - c)(1 - cq)} $$

so that

$$ \lim_{\alpha \to \infty} \hat{p}_n(x) = \lim_{\alpha \to \infty} \psi_n^{(\alpha, 2)}(x; 1/q), $$

which is the stated explicit formula in Theorem 4.1.

For the odd even-odd polynomials in Proposition 4.6, we choose $(a, b, c, d) = (cq^2\alpha, 1/\alpha, 1/\alpha, 1/\alpha)$,

$$ A_n(\alpha, q) = \frac{(1 - c\alpha q^{n+2})^3 (1 - cq^{n+1}/\alpha)}{\alpha cq^2(1 - cq^{2n+1})(1 - cq^{2n+2})}, $$

$$ C_n(\alpha, q) = \frac{\alpha cq^2(1 - \alpha q^n)(1 - q^{n-1}/\alpha)^3}{(1 - cq^{2n})(1 - cq^{2n+1})}. $$

As before choosing $\hat{p}_n(x) = 2^n\alpha^{-2n}B^{-n}p_n^{(\alpha)}(B\alpha^2x/2)$ and $B = -cq^2$ we find

$$ \lim_{\alpha \to \infty} \frac{1}{B\alpha^2} A_n(\alpha, q) = \frac{cq^{3n+2}}{(1 - cq^{2n+1})(1 - cq^{2n+2})} = \lambda_{2n+2}, $$

$$ \lim_{\alpha \to \infty} \frac{1}{B\alpha^2} C_n(\alpha, q) = \frac{q^n}{(1 - cq^{2n})(1 - cq^{2n+1})} = \lambda_{2n+1}. $$

The first degree limiting polynomial matches the first Ismail–Rahman solution in Theorem 4.4 with $(a, b, c, d) = (cq^2\alpha, 1/\alpha, 1/\alpha, 1/\alpha)$,

$$ x - \frac{1 + cq}{(1 - c)(1 - cq^2)} $$

so that

$$ \lim_{\alpha \to \infty} \hat{p}_n(x) = \lim_{\alpha \to \infty} \psi_n^{(\alpha, 1)}(x; q), $$

which is the stated explicit formula in Theorem 4.2.

We summarize these limits for the even-odd $q$-Lommel polynomials.
**Theorem 4.7.** The even-odd q-Lommel polynomials are the following limits of associated Askey–Wilson polynomials

$$p_{2n}(x; c, q) = \lim_{\alpha \to \infty} \frac{(2q)^n}{\alpha^{2n}} \psi_n^{(\alpha, 2)}(\alpha^2 x^2/2q; 1/q), \quad (a, b, c, d) = (\alpha/cq, 1/\alpha, 1/\alpha, 1/\alpha),$$

$$p_{2n+1}(x; c, q) = x \lim_{\alpha \to \infty} \frac{(-2/cq^2)^n}{\alpha^{2n}} \psi_n^{(\alpha, 1)}(-cq^2\alpha^2 x^2/2; q), \quad (a, b, c, d) = (cq^2\alpha, 1/\alpha, 1/\alpha, 1/\alpha).$$

For the classical q-Lommel polynomials $h_n(x; c; q)$, for the even polynomials choose

$$(a, b, c, d) = (1, q/\alpha^2, c, 1)$$

and for the odd polynomials choose

$$(a, b, c, d) = (1, q^2/\alpha^2, c, 1).$$

Similar calculations to the proof of Theorem 4.7 show the next result.

**Theorem 4.8.** The classical q-Lommel polynomials are the following limits of associated Askey–Wilson polynomials

$$h_{2n}(x; c, q) = \lim_{\alpha \to \infty} \frac{(-2)^n}{\alpha^{2n}} \psi_n^{(\alpha, 2)}(-\alpha^2 x^2/2; q), \quad (a, b, c, d) = (1, q/\alpha^2, c, 1),$$

$$h_{2n+1}(x; c, q) = x \lim_{\alpha \to \infty} \frac{(-2q)^n}{\alpha^{2n}} \psi_n^{(\alpha, 1)}(-\alpha^2 x^2/2q; q), \quad (a, b, c, d) = (1, q^2/\alpha^2, c, 1).$$

Theorem 4.9 is [15, Theorem 14.4.1].

**Theorem 4.9.** The classical q-Lommel polynomials are

$$h_n(x; c, q) = \sum_{k=0}^{n/2} \begin{bmatrix} n-k \\ k \end{bmatrix}_q \frac{(-c)^k q^{k^2-k}}{(c;q)_k (cq^{n-1}; q^{-1})_k} x^{n-2k}.$$

*Proof.* We consider the even case, the proof for the odd case is similar. The inner sum becomes an evaluable very-well-poised $_6W_5$

$${}_6W_5 \left( cq^{k-1}; q^k, cq^{n+k}, q^{k-n} \mid q; q^{-2k} \right) = \frac{(cq^{n-1}; q^{-1})_k (q^{n+1}; q)_k}{(c; q)_k (q^{k+1}; q)_k} q^{-k(n-k)}.$$

By considering the coefficient of $x^{2n-2k}$, we arrive at Theorem 4.9 with $n$ replaced by $2n$. The odd case actually gives the same result. $\square$

For the type $R_I$ q-Lommel polynomials there is a simple generating function which gives an explicit expression.

**Proposition 4.10.** The type $R_I$ q-Lommel polynomials have the generating function

$$\sum_{n=0}^{\infty} (c^{-1}; q^{-1})_n r_n(x; c, q) t^n = \sum_{k=0}^{\infty} \frac{(-xt/c)^k q^{-\binom{k}{2}}}{(t/c, tx; q^{-1})_{k+1}}.$$

*Proof.* If $G(x, t)$ is the generating function on the left side, then Definition 2.4 implies

$$G(x, t) - 1 = (x + 1/c)tG(x, t) - xt/c G(x, tq^{-1}) - xt^2/c G(x, t),$$

$$G(x, t) = \frac{1}{(1 - xt)(1 - t/c)} - \frac{xt/c}{(1 - xt)(1 - t/c)} G(x, tq^{-1})$$

whose iterate is the result. $\square$
**Theorem 4.11.** The type $R_I$ q-Lommel polynomials have the explicit formula

$$r_n(x; c, q) = \frac{1}{(c^{-1}; q^{-1})_n} \sum_{k=0}^{n} \sum_{a=0}^{n-k} (-x/c)^k q^{-\binom{k}{2}} \begin{bmatrix} k+a \\ a \end{bmatrix}_{q^{-1}} c^{-a} \begin{bmatrix} n-a \\ k \end{bmatrix}_{q^{-1}} x^{n-k-a}.$$

*Proof.* Apply the $q^{-1}$-binomial theorem to Proposition 4.10 to find the resulting coefficient of $t^n$. $\square$

**Proposition 4.12.** We have the connection coefficient relation

$$r_n(x^2; c, q) = \sum_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}_q \frac{c^k q^{n^2 - (n-k)^2}}{(cq^{n-1}, cq^{2n-k}; q^{-1})_k} p_{2n-2k}(x; c, q).$$

*Proof.* Induction on $n$ using the three-term relations. $\square$

**Proposition 4.13.** If $\mathcal{L}_p$ is the linear functional for the even-odd polynomials $p_n(x; c, q)$, then

$$\mathcal{L}_p(r_n(x^2; c, q)) = \frac{c^n q^{n^2}}{(c, cq; q)_n}.$$

*Proof.* Apply $\mathcal{L}_p$ to both sides of Proposition 4.12. By orthogonality, $\mathcal{L}_p(p_j(x)) = 0$ for $j > 0$, so only the $k = n$ term survives. $\square$

**Theorem 4.14.** The moments of the type $R_I$ q-Lommel polynomials are equal to the even moments of the even-odd q-Lommel polynomials,

$$\mathcal{L}_r(x^m) = \mathcal{L}_p(x^{2m}), \quad m \ge 0.$$

*Proof.* Using the three-term recurrence (2.1), the type $R_I$ moments $\mathcal{L}_r(x^m)$ are recursively determined by [18, Corollary 3.15]

$$\mathcal{L}_r(r_n(x; c, q)) = a_1 a_2 \cdots a_n = \frac{c^n q^{n^2}}{(c, cq; q)_n}, \quad n \ge 0.$$

By Proposition 4.13 the moments $\mathcal{L}_p(x^{2m})$ satisfy the same recurrence. $\square$

For completeness, we give the inverse relation to Proposition 4.12.

**Proposition 4.15.** We have the connection coefficient relation

$$p_{2n}(x; c, q) = \sum_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}_q \frac{(-c)^k q^{2nk - \binom{k+1}{2}}}{(cq^{n-1}, cq^{2n-1}; q^{-1})_k} r_{n-k}(x^2; c, q).$$

**Proposition 4.16.** The even-odd q-Lommel polynomials have the explicit expressions

$$p_{2n}(x; c, q) = \frac{1}{(c; q)_{2n}} \sum_{k=0}^{n} (-1)^k x^{2n-2k} (cq^k; q)_{2n-2k} \sum_{j=0}^{k} \begin{bmatrix} n-j \\ k-j \end{bmatrix}_q \begin{bmatrix} n-k+j-1 \\ j \end{bmatrix}_q c^j q^{jn+\binom{k}{2}},$$

$$p_{2n+1}(x; c, q) = \frac{1}{(c; q)_{2n+1}} \sum_{k=0}^{n} (-1)^k x^{2n-2k+1} (cq^k; q)_{2n-2k+1} \sum_{j=0}^{k} \begin{bmatrix} n-j \\ k-j \end{bmatrix}_q \begin{bmatrix} n-k+j \\ j \end{bmatrix}_q c^j q^{jn+\binom{k}{2}}.$$

*Proof.* This follows from Definition 2.3, by considering the coefficients of $x^{2n-2k-1}$. $\square$
## 5. MOMENTS AND CONTINUED FRACTIONS

In this section we review the known facts which connect continued fractions to moment generating functions. We independently prove the continued fractions for the moment generating functions of each of the three q-Lommel polynomials.

**Definition 5.1.** Take a sequence of orthogonal polynomials $p_n(x)$ which satisfy $p_{-1}(x) = 0$, $p_0(x) = 1$, and

$$p_{n+1}(x) = (x - b_n)p_n(x) - \lambda_n p_{n-1}(x), \quad n \ge 0,$$

and whose linear functional for orthogonality is $\mathcal{L}_p$. Define

$$\mu_n(\{b_k\}_{k \ge 0}, \{\lambda_k\}_{k \ge 0}) = \mathcal{L}_p(x^n).$$

The moment generating function for $\mathcal{L}_p$ is

$$\sum_{n=0}^{\infty} \mathcal{L}_p(x^n) t^n = \sum_{n=0}^{\infty} \mu_n(\{b_k\}_{k \ge 0}, \{\lambda_k\}_{k \ge 0}) t^n.$$

A Jacobi continued fraction also exists, converging as formal power series in $t$:

$$\sum_{n=0}^{\infty} \mathcal{L}_p(x^n) t^n = \frac{1}{1 - b_0 t - \frac{\lambda_1 t^2}{1 - b_1 t - \frac{\lambda_2 t^2}{1 - \ddots}}}, \quad (5.1)$$

**Definition 5.2 ([18]).** For general type $R_I$ orthogonal polynomials

$$r_{n+1}(x) = (x - b_n)r_n(x) - (a_n x + \lambda_n)r_{n-1}(x), \quad n \ge 0,$$

with linear functional $\mathcal{L}_r$, define

$$\mu_n(\{b_k\}_{k \ge 0}, \{a_k\}_{k \ge 0}, \{\lambda_k\}_{k \ge 0}) = \mathcal{L}_r(x^n). \quad (5.2)$$

The corresponding continued fraction for the type $R_I$ moment generating function is [18, Corollary 3.7]

$$\sum_{n=0}^{\infty} \mathcal{L}_r(x^n) t^n = \frac{1}{1 - b_0 t - \frac{a_1 t + \lambda_1 t^2}{1 - b_1 t - \frac{a_2 t + \lambda_2 t^2}{1 - \ddots}}}, \quad (5.3)$$

Note that both continued fractions in (5.1) and (5.3) are explicitly given in terms of the three-term recurrence coefficients. We shall evaluate the continued fractions as quotients of basic hypergeometric series, namely q-Bessel functions, using contiguous relations.

For the Lommel polynomials $h_n(x; c)$, it is known that the moment generating function is a quotient of Bessel functions, with $\lambda_n = 1/(c+n-1)(c+n)$,

$$\sum_{n=0}^{\infty} \mathcal{L}_h(x^n) t^n = \frac{\binom{c+1}{0} F_1(c+1; -t^2)}{\binom{c}{0} F_1(c; -t^2)} = \frac{1}{1 - \frac{\lambda_1 t^2}{1 - \frac{\lambda_2 t^2}{1 - \ddots}}},$$
The moment generating function for the classical q-Lommel polynomials is a quotient of q-Bessel functions. In this section we shall see that a corresponding result holds for our other two q-Lommel polynomials, and in fact their moment generating functions are equal.

**Theorem 5.3** ([15, Theorem 14.4.3]). *The moment generating function for the classical q-Lommel polynomials $h_n(x; c, q)$ is a quotient of Jackson's first q-Bessel functions*

$$ \sum_{n=0}^{\infty} \mathcal{L}_h(x^n) t^n = \frac{2\phi_1(0, 0; cq; q, -t^2)}{2\phi_1(0, 0; c; q; -t^2)} = \frac{1}{1 - \frac{\lambda_1 t^2}{1 - \frac{\lambda_2 t^2}{1 - \ddots}}} $$

where, as in Definition 2.2, $\lambda_n = cq^{n-1}/((1 - cq^{n-1})(1 - cq^n))$.

**Theorem 5.4.** *The moment generating function for the even-odd q-Lommel polynomials $p_n(x; c, q)$ is a quotient of Jackson's third q-Bessel functions*

$$ \sum_{n=0}^{\infty} \mathcal{L}_p(x^n) t^n = \frac{1}{1 - \frac{\lambda_1 t^2}{1 - \frac{\lambda_2 t^2}{1 - \ddots}}} $$

where, as in Definition 2.3,

$$ \lambda_{2n} = \frac{cq^{3n-1}}{(1 - cq^{2n-1})(1 - cq^{2n})}, \quad \lambda_{2n+1} = \frac{q^n}{(1 - cq^{2n})(1 - cq^{2n+1})} $$

**Theorem 5.5.** *The moment generating function for the type $R_I$ q-Lommel polynomials $r_n(x; c, q)$ is a quotient of Jackson's third q-Bessel functions*

$$ \sum_{n=0}^{\infty} \mathcal{L}_r(x^n) z^n = \frac{1}{1 - b_0 z - \frac{a_1 z}{1 - b_1 z - \frac{a_2 z}{1 - b_2 z - \dots}}} $$

where, as in Definition 2.4,

$$ a_n = \frac{cq^{2n-1}}{(1 - cq^{n-1})(1 - cq^n)}, \quad b_n = \frac{q^n}{1 - cq^n}. $$

Theorem 4.14 implies that the two continued fractions in Theorems 5.4 and 5.5 with $z = t^2$ are equal.

**Corollary 5.6.** *We have the equality of continued fractions*

$$ \frac{1}{1 - b_0 z - \frac{a_1 z}{1 - b_1 z - \frac{a_2 z}{1 - b_2 z - \dots}}} = \frac{1}{1 - \frac{\lambda_1 z}{1 - \frac{\lambda_2 z}{1 - \dots}}} $$

where $a_n$, $b_n$, and $\lambda_n$ are defined as in Theorems 5.4 and 5.5.
Theorems 5.3, 5.4, and 5.5 may all be proven using contiguous relations for hypergeometric and basic hypergeometric series. To prove Theorems 5.3 and 5.4 we use Heine's contiguous relation [10, Eq. 17.6.19] which is

$$ {}_2\phi_1(aq, b; cq; q, z) - {}_2\phi_1(a, b; c; q, z) = \frac{(1-b)(a-c)z}{(1-c)(1-cq)} {}_2\phi_1(aq, bq; cq^2; q, z). $$

Equivalently,

$$ \frac{{}_2\phi_1(aq, b; cq; q, z)}{{}_2\phi_1(a, b; c; q, z)} = \frac{1}{1 - \frac{(1-b)(a-c)z} {(1-c)(1-cq)} \cdot \frac{{}_2\phi_1(bq, aq; cq^2; q, z)}{{}_2\phi_1(b, aq; cq; q, z)}} \quad (5.4) $$

Applying (5.4) iteratively, we obtain Heine's continued fraction, which is a q-analogue of Gauss's continued fraction.

**Lemma 5.7 (Heine's fraction).** We have

$$ \frac{{}_2\phi_1(aq, b; cq; q, z)}{{}_2\phi_1(a, b; c; q, z)} = \frac{1}{1 - \frac{\beta_1 z}{1 - \frac{\beta_2 z}{1 - \ddots}}} $$

where

$$ \beta_{2n+1} = \frac{(1 - b q^n)(a - c q^n) q^n}{(1 - c q^{2n})(1 - c q^{2n+1})}, $$

$$ \beta_{2n} = \frac{(1 - a q^n)(b - c q^n) q^{n-1}}{(1 - c q^{2n-1})(1 - c q^{2n})}. $$

Theorem 5.3 is the special case $a = b = 0$ and $z = -t^2$ of Lemma 5.7. Theorem 5.4 is also the limiting case $z = -t^2/a$, $b = 0$, $a \to \infty$ of Lemma 5.7.

For Theorem 5.5 we need the q-Nörlund fraction [8, Eq. (19.2.7)]. However, to simplify the expressions we need some notation for continued fractions.

**Definition 5.8.** For sequences $a_i$ and $b_i$, let

$$ \mathbf{K}_{i=0}^{m} \left( \frac{a_i}{b_i} \right) = \frac{a_0}{b_0 + \frac{a_1}{b_1 + \ddots + \frac{a_m}{b_m}}} $$

$$ \mathbf{K}_{i=0}^{\infty} \left( \frac{a_i}{b_i} \right) = \frac{a_0}{b_0 + \frac{a_1}{b_1 + \ddots}} $$

The following lemma will be used later.

**Lemma 5.9.** For any sequences $\{a_i : 0 \le i \le m\}$, $\{b_i : 0 \le i \le m\}$, and $\{c_i : -1 \le i \le m\}$, we have

$$ \mathbf{K}_{i=0}^{m} \left( \frac{a_i}{b_i} \right) = \frac{1}{c_{-1}} \mathbf{K}_{i=0}^{m} \left( \frac{a_i c_{i-1} c_i}{b_i c_i} \right). $$

*Proof.* By multiplying the numerator and denominator of the $i^{th}$ fraction by $c_i$, we obtain

$$ \frac{a_0}{b_0 + \frac{a_1}{b_1 + \dots + \frac{a_m}{b_m}}} = \frac{a_0 c_0}{b_0 c_0 + \frac{a_1 c_0 c_1}{b_1 c_1 + \dots + \frac{a_m c_{m-1} c_m}{b_m c_m}}} $$

which is equivalent to the equation in the lemma.

□
**Lemma 5.10** (q-Nörlund fraction). We have

$$ \frac{_{2}\phi_{1}(a, b; c; q, z)}{_{2}\phi_{1}(aq, bq; cq; q, z)} = \frac{1 - c - (a + b - ab - abq)z}{1 - c} + \frac{1}{1 - c} \sum_{m=1}^{\infty} \mathbf{K} \left( \frac{c_m(z)}{e_m + d_m z} \right), $$

where

$$ \begin{aligned} c_m(z) &= (1 - a q^m)(1 - b q^m)(cz - abq^m z^2)q^{m-1}, \\ e_m &= 1 - c q^m, \\ d_m &= -(a + b - abq^m - abq^{m+1})q^m. \end{aligned} $$

The q-Nörlund fraction can be restated in the form of a continued fraction for type $R_I$ orthogonal polynomials.

**Proposition 5.11** (q-Nörlund fraction restated). We have

$$ \frac{_{2}\phi_{1}(aq, bq; cq; q, z)}{_{2}\phi_{1}(a, b; c; q, z)} = \frac{1}{1 - b_{0}z - \frac{a_{1}z + \lambda_{1}z^{2}}{1 - b_{1}z - \frac{a_{2}z + \lambda_{2}z^{2}}{1 - b_{2}z - \ddots}}}, $$

where

$$ \begin{aligned} b_m &= \frac{(a + b - abq^m - abq^{m+1})q^m}{1 - c q^m}, \\ a_m &= \frac{(1 - a q^m)(1 - b q^m) c q^{m-1}}{(1 - c q^{m-1})(1 - c q^m)}, \\ \lambda_m &= \frac{(1 - a q^m)(1 - b q^m) a b q^{2m-1}}{(1 - c q^{m-1})(1 - c q^m)}. \end{aligned} $$

*Proof.* By taking the inverse on each side of the equation in Lemma 5.10, we obtain

$$ \frac{_{2}\phi_{1}(aq, bq; cq; q, z)}{_{2}\phi_{1}(a, b; c; q, z)} = \frac{1 - c}{c_{0}(z)} \sum_{m=0}^{\infty} \mathbf{K} \left( \frac{c_{m}(z)}{e_{m} + d_{m}z} \right). $$

Applying Lemma 5.9 with $c_i = 1/(1 - c q^i)$ and $m \to \infty$ yields

$$ \frac{_{2}\phi_{1}(aq, bq; cq; q, z)}{_{2}\phi_{1}(a, b; c; q, z)} = \frac{(1 - c q^{-1})(1 - c)}{c_{0}(z)} \sum_{m=0}^{\infty} \mathbf{K} \left( \frac{c_{m}(z)/(1 - c q^{m-1})(1 - c q^{m})}{e_{m}/(1 - c q^{m}) + d_{m}z/(1 - c q^{m})} \right), $$

which is the same as the desired identity.

*Proof of Theorem 5.5.* Replace $z$ by $z/b$, put $a = 0$, and let $b \to \infty$ in Proposition 5.11. The result is Theorem 5.5. □

Note that when $b = 0$ both Lemma 5.7 and Proposition 5.11 give a continued fraction expression for

$$ \frac{_{2}\phi_{1}(a, 0; c; q, z)}{_{2}\phi_{1}(aq, 0; cq; q, z)}. $$

Therefore we obtain the following theorem.
**Theorem 5.12.** We have the equality of continued fractions

$$ \frac{1}{1 - b_0 z - \frac{a_1 z}{1 - b_1 z - \frac{a_2 z}{1 - b_2 z - \ddots}}} = \frac{1}{1 - \frac{\lambda_1 z}{1 - \frac{\lambda_2 z}{1 - \ddots}}} $$

where

$$ a_n = \frac{(aq^n - 1)cq^{n-1}}{(1 - cq^{n-1})(1 - cq^n)}, \quad b_n = \frac{aq^n}{1 - cq^n}, \\ \lambda_{2n} = \frac{-cq^{2n-1}(1 - aq^n)}{(1 - cq^{2n-1})(1 - cq^{2n})}, \quad \lambda_{2n+1} = \frac{(a - cq^n)q^n}{(1 - cq^{2n})(1 - cq^{2n+1})}. $$

When Theorem 5.12 is interpreted as an equality for moment generating functions, we find the following generalization of Theorem 4.14 which holds for q-Lommel polynomials.

**Corollary 5.13.** Let $\lambda_n$, $a_n$ and $b_n$ be given by Theorem 5.12. The $2n^{th}$ moment of the orthogonal polynomials defined by $p_{n+1}(x) = xp_n(x) - \lambda_n p_{n-1}(x)$ is equal to the $n^{th}$ moment of the type $R_I$ polynomials defined by $r_{n+1}(x) = (x - b_n)r_n(x) - a_n x r_{n-1}(x)$.

## 6. COMBINATORICS OF MOMENTS OF TYPE $R_I$ q-LOMMEL POLYNOMIALS

The moment generating function for type $R_I$ polynomials is given by the continued fraction in (5.3). For type $R_I$ q-Lommel polynomials we give in this section a general combinatorial interpretation for this infinite continued fraction in terms of parallelogram polyominoes. We also interpret the finite continued fraction and give an explicit rational expression using q-Lommel polynomials. To be specific we give a combinatorial interpretation for the ratio

$$ r_n^{(3)}(x^{-1}; q^{\nu+2}, q) / r_{n+1}^{(3)}(x^{-1}; q^{\nu+1}, q) $$

of (rescaled) type $R_I$ q-Lommel polynomials, Theorem 6.9. This is a finite version of the result of Bousquet-Mélou and Viennot [4]. The $n \to \infty$ limit of Theorem 6.9 yields a quotient of q-Bessel functions,

$$ J_{\nu+1}^{(3)}(x^{1/2}; q^{-1}) / J_{\nu}^{(3)}(x^{1/2}; q^{-1}) $$

which is the moment generating function for the type $R_I$ q-Lommel polynomials. This material appears in our unpublished manuscript [17, Section 5].

We shall need several definitions related to parallelogram polyominoes and Motzkin paths.

**Definition 6.1.** An NE-path is a lattice path from $(0, 0)$ to $(a, b)$ for some positive integers $a, b$ consisting of north steps $(0, 1)$ and east steps $(1, 0)$. A parallelogram polyomino is a set of unit squares enclosed by two NE-paths with the same ending points that do not intersect except the starting and ending points. Denote by $\mathcal{P}$ the set of parallelogram polyominoes.

For a parallelogram polyomino $\alpha \in \mathcal{P}$ let $U(\alpha)$ be the upper boundary path and $D(\alpha)$ the lower boundary path; see Figure 1. A diagonal of $\alpha$ is the set of squares in $\alpha$ whose centers are on the line $x + y = i$ for some integer $i$. The size of a diagonal is the number of squares in it. See Figure 2.
<figure><img src="image_101.png" /><figcaption>FIGURE 1. The boundary paths $U(\alpha)$ and $D(\alpha)$ for a parallelogram polyomino.</figcaption></figure>

<figure><img src="image_102.png" /><figcaption>FIGURE 2. A diagonal with size 3 in a parallelogram polyomino.</figcaption></figure>

<figure><img src="image_103.png" /><figcaption>FIGURE 3. From left to right are shown an NN-diagonal, EE-diagonal, NE-diagonal, and EN-diagonal of size $n+1$ whose weights are, respectively, $a_n, b_n, c_n$, and $d_n$.</figcaption></figure>

**Definition 6.2.** We denote by $P^{\le k}$ the set of parallelogram polyominoes in which every diagonal has size at most $k$.

Consider $\alpha \in P$ and a diagonal $\tau$ of $\alpha$. Let $u$ (resp. $d$) be the northwest (resp. southeast) corner of the topmost (resp. bottommost) square of $\tau$. There are four cases for $d$. We say that $d$ is an *NN-diagonal* (resp. *NE-diagonal*, *EN-diagonal*, and *EE-diagonal*) if the step in $U(\alpha)$ starting at $u$ is a north (resp. north, east, and east) step and the step in $D(\alpha)$ starting at $d$ is a north (resp. east, north, and east) step. See Figure 3.

For sequences $\{a_n\}_{n \ge 0}, \{b_n\}_{n \ge 0}, \{c_n\}_{n \ge 0}$, and $\{d_n\}_{n \ge 0}$, define the *weight* $\text{wt}(\alpha; a, b, c, d)$ of $\alpha \in P$ to be the product of $a_n$ (resp. $b_n, c_n$, and $d_n$) for each NN-diagonal (resp. EE-diagonal, NE-diagonal, and EN-diagonal) of size $n+1$.

Now we review Flajolet's theory [13] on continued fraction expressions for Motzkin path generating functions.

**Definition 6.3.** A Motzkin path is a lattice path from (0, 0) to (n, 0) consisting of up steps (1, 1), down steps (1, -1), and horizontal steps (1, 0) that never goes below the x-axis. A 2-Motzkin path is a Motzkin path in which every horizontal step is colored red or blue. The height of a 2-Motzkin path is the largest integer y for which (x, y) is a point in the path.
<figure><img src="image_104.png" /><figcaption>FIGURE 4. A 2-Motzkin path $p$ in $\text{Motz}_2^{\le 3}$ with $\text{wt}(p; a, b, c, d) = a_2^2 b_0 b_1 b_2 c_0 c_1^2 c_2 d_1 d_2^2 d_3$. The blue horizontal edges are represented by double edges.</figcaption></figure>

Denote by $\text{Motz}_2$ the set of all 2-Motzkin paths and by $\text{Motz}_2^{\le m}$ the set of all 2-Motzkin paths with height at most $m$.

For sequences $\{a_n\}_{n \ge 0}$, $\{b_n\}_{n \ge 0}$, $\{c_n\}_{n \ge 0}$, and $\{d_n\}_{n \ge 0}$, define the weight $\text{wt}(p; a, b, c, d)$ of a 2-Motzkin path $p$ to be the product of $a_n$ (resp. $b_n$, $c_n$, and $d_n$) for each red horizontal step (resp. blue horizontal step, up step, and down step) starting at height $n$; see Figure 4.

Flajolet's theory [13] proves the following lemma for a finite continued fraction.

**Lemma 6.4.** Given sequences $\{a_n\}_{n \ge 0}$, $\{b_n\}_{n \ge 0}$, $\{c_n\}_{n \ge 0}$, and $\{d_n\}_{n \ge 0}$, we have

$$ \sum_{p \in \text{Motz}_2^{\le m}} \text{wt}(p; a, b, c, d) = \frac{1}{1 - a_0 - b_0 - \frac{c_0 d_1}{1 - a_1 - b_1 - \cdots - \frac{c_{m-1} d_m}{1 - a_m - b_m}}} $$

There is a well-known bijection between 2-Motzkin paths and parallelogram polyominoes.

**Definition 6.5** (The map $\phi: \text{Motz}_2^{\le m} \to \mathcal{P}^{\le m+1}$). Let $p \in \text{Motz}_2^{\le m}$. Then $\phi(p) = \alpha$ is the parallelogram polyomino whose upper and lower boundary paths $U, D$ are constructed by the following algorithm.

(1) The first step of $U$ (resp. $D$) is a north (resp. east) step.

(2) For $i = 1, 2, \dots, n$, where $n$ is the number of steps in $p$, the $(i+1)^{\text{st}}$ steps of $U$ and $D$ are defined as follows.

(a) If the $i^{\text{th}}$ step of $p$ is an up step, then the $(i+1)^{\text{st}}$ step of $U$ (resp. $D$) is a north (resp. east) step.

(b) If the $i^{\text{th}}$ step of $p$ is a down step, then the $(i+1)^{\text{st}}$ step of $U$ (resp. $D$) is a east (resp. north) step.

(c) If the $i^{\text{th}}$ step of $p$ is a red horizontal step, then the $(i+1)^{\text{st}}$ steps of $U$ and $D$ are both north steps.

(d) If the $i^{\text{th}}$ step of $p$ is a blue horizontal step, then the $(i+1)^{\text{st}}$ steps of $U$ and $D$ are both east steps.

(3) Finally, the last step of $U$ (resp. $D$) is an east (resp. north) step.

For example, if $p$ is the 2-Motzkin path in Figure 4, then $\phi(p)$ is the parallelogram polyomino $\alpha$ in Figure 1.

It is easy to see from the construction that $\phi: \text{Motz}_2^{\le m} \to \mathcal{P}^{\le m+1}$ is a bijection such that if $\phi(p) = \alpha$, then $\text{wt}(\alpha; a, b, c, d) = d_0 \text{wt}(p; a, b, c, d)$.

Therefore we obtain the following proposition from Lemma 6.4, which changes the weighted 2-Motzkin paths into weighted parallelogram polyominoes.
**Proposition 6.6.** Given sequences ${a_n}_{n \ge 0}$, ${b_n}_{n \ge 0}$, ${c_n}_{n \ge 0}$, and ${d_n}_{n \ge 0}$, we have

$$ \sum_{\alpha \in \mathcal{P} \leq m+1} \mathrm{wt}(\alpha; a, b, c, d) = \frac{d_0}{1 - a_0 - b_0 - \frac{c_0 d_1}{1 - a_1 - b_1 - \cdots - \frac{c_{m-1} d_m}{1 - a_m - b_m}}} $$

As a special case in Proposition 6.6, if ${a_n}_{n \ge 0}$, ${b_n}_{n \ge 0}$, ${c_n}_{n \ge 0}$, and ${d_n}_{n \ge 0}$ are the sequences given by $a_n = q^{n+1}Y$, $b_n = q^{n+1}X$, $c_n = q^{n+1}XY$, and $d_n = q^{n+1}$, then one can easily check that

$$ XY \cdot \mathrm{wt}(\alpha; a, b, c, d) = X^{\mathrm{col}(\alpha)} Y^{\mathrm{row}(\alpha)} q^{\mathrm{area}(\alpha)}. $$

Thus we obtain the following corollary.

**Corollary 6.7.** We have

$$ \sum_{\alpha \in \mathcal{P} \leq m+1} X^{\mathrm{col}(\alpha)} Y^{\mathrm{row}(\alpha)} q^{\mathrm{area}(\alpha)} = \frac{qXY}{1 - q(X + Y) - \frac{q^3 XY}{1 - q^2(X + Y) - \cdots - \frac{q^{2m+1} XY}{1 - q^{m+1}(X + Y)}}} $$

For the rest of this section we will find a finite version of the following result due to Bousquet-Mélou and Viennot [4].

**Theorem 6.8** ([9] for $\nu = 0$ and [4] for general $\nu$). The trivariate generating function for parallelogram polyominoes is

$$ \sum_{\alpha \in \mathcal{P}} (q^{\nu} x)^{\mathrm{col}(\alpha)} (q^{\nu})^{\mathrm{row}(\alpha)} q^{\mathrm{area}(\alpha)} = -q^{\nu} x^{1/2} \frac{J_{\nu+1}^{(3)}(x^{1/2}; q^{-1})}{J_{\nu}^{(3)}(x^{1/2}; q^{-1})}. $$

In fact Delest and Fédou [9] (for $\nu = 0$), and Bousquet-Mélou and Viennot [4] state their results in the following equivalent form:

$$ \sum_{\alpha \in \mathcal{P}} x^{\mathrm{col}(\alpha)} y^{\mathrm{row}(\alpha)} q^{\mathrm{area}(\alpha)} = \frac{qxy}{1 - qy} \cdot \frac{1\phi_1(0; q^2y; q, q^2x)}{1\phi_1(0; qy; q, qx)}. $$

Bousquet-Mélou and Viennot [4] also showed that

$$ \sum_{\alpha \in \mathcal{P}} x^{\mathrm{col}(\alpha)} y^{\mathrm{row}(\alpha)} q^{\mathrm{area}(\alpha)} = \frac{qxy}{1 - q(x + y) - \frac{q^3xy}{1 - q^2(x + y) - \frac{q^5xy}{\ddots}}} \quad (6.1) $$

We note that in [4, Corollary 4.6] the sequence of the coefficients of $(x + y)$ in the continued fraction (6.1) was inadvertently written $q, q^3, q^5, \dots$, where the correct sequence is $q, q^2, q^3, \dots$. We also note that there are similar results in [1].

For a sequence $s = \{s_n\}_{n \ge 0}$, define $\delta s = \{s_{n+1}\}_{n \ge 0}$. Kim and Stanton [18, Eq. (5.4)] showed that for given sequences $b = \{b_n\}_{n \ge 0}$, $a = \{a_n\}_{n \ge 0}$, and $\lambda = \{\lambda_n\}_{n \ge 0}$, and for a nonnegative integer $k$,

$$ \frac{x^m P_m(x^{-1}; \delta b, \delta a, \delta \lambda)}{x^{m+1} P_{m+1}(x^{-1}; b, a, \lambda)} = \frac{1}{-a_0 x - \lambda_0 x^2} \mathbf{K}_{i=0}^{m} \left( \frac{-a_i x - \lambda_i x^2}{1 - b_i x} \right). \quad (6.2) $$
Now we are ready to prove a finite version of Theorem 6.8.

**Theorem 6.9.** The trivariate generating function for bounded diagonal parallelogram polyominoes is

$$ \sum_{\alpha \in \mathcal{P}^{\le m+1}} (q^{\nu} x)^{\text{col}(\alpha)} (q^{\nu})^{\text{row}(\alpha)} q^{\text{area}(\alpha)} = \frac{q^{2\nu+1}}{1-q^{\nu+1}} \cdot \frac{r_m^{(3)}(x^{-1}; q^{\nu+2}, q)}{r_{m+1}^{(3)}(x^{-1}; q^{\nu+1}, q)}. \quad (6.3) $$

*Proof.* Let $b = \{b_i\}_{i \ge 0}$, $a = \{a_i\}_{i \ge 0}$, and $\lambda = \{\lambda_i\}_{i \ge 0}$, where

$$ b_i = \frac{q^{\nu+i+1}}{1-q^{\nu+i+1}}, \quad a_i = \frac{q^{2\nu+2i+1}}{(1-q^{\nu+i})(1-q^{\nu+i+1})}, \quad \lambda_i = 0. $$

Then $P_m(x; b, a, \lambda) = r_m^{(3)}(x; q^{\nu+1}, q)$ and $P_m(x; \delta b, \delta a, \delta \lambda) = r_m^{(3)}(x; q^{\nu+2}, q)$. By (6.2),

$$ \frac{r_m^{(3)}(x^{-1}; q^{\nu+2}, q)}{x r_{m+1}^{(3)}(x^{-1}; q^{\nu+1}, q)} = \frac{x^m P_m(x^{-1}; \delta b, \delta a, \delta \lambda)}{x^{m+1} P_{m+1}(x^{-1}; b, a, \lambda)} = \frac{1}{-a_0 x} \mathbf{K}_{i=0}^{m} \left( \frac{-a_i x}{1-b_i x} \right). $$

By Lemma 5.9 with $c_i = 1 - q^{\nu+i+1}$,

$$ \begin{aligned} \frac{1}{-a_0 x} \mathbf{K}_{i=0}^{m} \left( \frac{-a_i x}{1-b_i x} \right) &= \frac{1}{-a_0 x} \frac{1}{c_{-1}} \mathbf{K}_{i=0}^{m} \left( \frac{-a_i c_{i-1} c_i x}{c_i - b_i c_i x} \right) \\ &= \frac{1 - q^{\nu+1}}{-q^{2\nu+1} x} \mathbf{K}_{i=0}^{m} \left( \frac{-q^{2\nu+2i+1} x}{1 - q^{\nu+i+1} - q^{\nu+i+1} x} \right). \end{aligned} $$

Letting $X = q^{\nu}x$ and $Y = q^{\nu}$, and combining the above equations, we obtain

$$ \frac{q^{2\nu+1}}{1-q^{\nu+1}} \cdot \frac{r_m^{(3)}(x^{-1}; q^{\nu+2}, q)}{r_{m+1}^{(3)}(x^{-1}; q^{\nu+1}, q)} = - \mathbf{K}_{i=0}^{m} \left( \frac{-q^{2i+1}XY}{1-q^{i+1}(X+Y)} \right). $$

Corollary 6.7 then completes the proof. $\square$

<figure><img src="image_105.png" /></figure>

*Remark 6.10.* It is well known that the generating function for bounded Motzkin paths is given by a ratio of orthogonal polynomials; see e.g. Flajolet [13], Viennot [28, Ch. V, Eq. (27)], or Krattenthaler [24, Theorem 10.11.1]. The argument in this section implies that the left-hand side of (6.3) is the generating function for certain weighted bounded Motzkin paths. Theorem 6.9 shows that this generating function is also equal to a ratio of type $R_I$ polynomials.

By (3.4), taking the limit $m \to \infty$ in Theorem 6.9 we obtain Theorem 6.8. We may also use Theorem 4.11 to write the finite continued fraction as an explicit rational function.

**Corollary 6.11.** The trivariate generating function for bounded diagonal parallelogram polyominoes is

$$ \sum_{\alpha \in \mathcal{P}^{\le n+1}} x^{\text{col}(\alpha)} y^{\text{row}(\alpha)} q^{\text{area}(\alpha)} = - \frac{x \sum_{k=0}^{n} \sum_{a=0}^{n-k} (-1)^k x^a y^{-k-a} q^{-\binom{k}{2}-2k} \begin{bmatrix} k+a \\ a \end{bmatrix}_{q^{-1}} \begin{bmatrix} n-a \\ k \end{bmatrix}_{q^{-1}}}{\sum_{k=0}^{n+1} \sum_{a=0}^{n+1-k} (-1)^k x^a y^{-k-a} q^{-\binom{k}{2}-k} \begin{bmatrix} k+a \\ a \end{bmatrix}_{q^{-1}} \begin{bmatrix} n+1-a \\ k \end{bmatrix}_{q^{-1}}}. $$

Cigler and Krattenthaler [7] found a different finite version of Theorem 6.8.
**Theorem 6.12** ([7, Corollary 55]). For any integer $k \ge 1$, we have

$$ \sum_{\alpha \in \mathcal{P}_1^{\le k}} x^{\mathrm{col}(\alpha)} y^{\mathrm{row}(\alpha)} q^{\mathrm{area}(\alpha)} = - \frac{y \sum_{j=1}^{k} (-1)^j x^j q^{\binom{j+1}{2}} \sum_{i=0}^{k-j} (yq)^i \left[ \begin{smallmatrix} k-i-1 \\ j-1 \end{smallmatrix} \right]_q \left[ \begin{smallmatrix} i+j-1 \\ j-1 \end{smallmatrix} \right]_q}{\sum_{j=0}^{k} (-1)^j x^j q^{\binom{j+1}{2}} \sum_{i=0}^{k-j} (yq)^i \left[ \begin{smallmatrix} k-i \\ j \end{smallmatrix} \right]_q \left[ \begin{smallmatrix} i+j-1 \\ j-1 \end{smallmatrix} \right]_q}, $$

where $\mathcal{P}_1^{\le k}$ is the set of parallelogram polyominoes such that each column has length at most $k$ and $\left[ \begin{smallmatrix} i \\ -1 \end{smallmatrix} \right]_q = \delta_{i,-1}$.

**Remark 6.13.** The second odd-even trick (7.1) with $\lambda_{2k-1} = q^k y$ and $\lambda_{2k} = q^k$ gives

$$ 1 + \frac{qy}{1 - q(x + y) - \frac{q^3 xy}{1 - q^2(x + y) - \frac{q^5 xy}{\dots}}} = \frac{1}{1 - \frac{qy}{1 - \frac{qx}{1 - \frac{q^2 y}{1 - \frac{q^2 x}{\dots}}}}} $$

**Remark 6.14.** There are also finite versions of Theorem 6.9 for the classical $q$-Lommel polynomials and the even-odd $q$-Lommel polynomials. The rational function is again a quotient of orthogonal polynomials while the weights on $\mathcal{P}^{\le m+1}$ depend upon the diagonals.

Here are the infinite continued fractions for these two cases. For the classical $q$-Lommel polynomials, Theorem 5.3 becomes

$$ \frac{2\phi_1(0, 0; q^2y; q; -qx)}{2\phi_1(0, 0; qy; q; -qx)} = \frac{1 - qy}{1 - qy - \frac{q^2xy}{1 - q^2y - \frac{q^3xy}{1 - q^3y - \frac{q^4xy}{\dots}}}} $$

For the even-odd $q$-Lommel polynomials, Theorem 5.4 becomes

$$ \frac{1\phi_1(0; q^2y; q; q^2x)}{1\phi_1(0; qy; q; qx)} = \frac{1 - qy}{1 - qy - \frac{A_1}{1 - q^2y - \frac{A_2}{1 - q^3y - \frac{A_3}{\dots}}}} $$

where $A_{2k-1} = xq^k$ and $A_{2k} = xyq^{3k/2+1}$.

## 7. CONCURRENCE OF MOMENTS

Recall the notation (5.2) for the moments $\mu_n(\{b_k\}_{k \ge 0}, \{a_k\}_{k \ge 0}, \{\lambda_k\}_{k \ge 0})$, which we also write $\mu_n(\{b_k\}, \{a_k\}, \{\lambda_k\})$, or $\mu_n(\{b_k\}, \{\lambda_k\})$ when $a_k = 0$. There is a concurrence of moments (see Propositions 4.5 and 4.6), which we call the first and second odd-even tricks

$$ \begin{aligned} \mu_{2n}(\{0\}, \{\lambda_k\}) &= \mu_n(\{\lambda_{2k} + \lambda_{2k+1}\}, \{\lambda_{2k}\lambda_{2k-1}\}), \\ \mu_{2n+2}(\{0\}, \{\lambda_k\}) &= \lambda_1 \mu_n(\{\lambda_{2k+2} + \lambda_{2k+1}\}, \{\lambda_{2k}\lambda_{2k+1}\}). \end{aligned} \quad (7.1) $$
The classical orthogonal polynomial moments are a special case of type $R_I$ moments

$$ \mu_n(\{b_k\}, \{0\}, \{\lambda_k\}) = \mu_n(\{b_k\}, \{\lambda_k\}). $$

There is another concurrence of moments, which follows from [18, Corollary 3.7]

$$ \mu_{2n}(\{0\}, \{a_k\}) = \mu_n(\{0\}, \{a_k\}, \{0\}). \qquad (7.2) $$

It is known [18] that a type $R_I$ moment $\mu_n(\{b_k\}, \{a_k\}, \{\lambda_k\})$ is a nonnegative polynomial in the recurrence coefficients. Besides (7.2) Theorem 4.14 is another example of classical orthogonal polynomial moments being equal to type $R_I$ moments

$$ \mu_{2n}(\{0\}, \{\Lambda_k\}) = \mu_n(\{b_k\}, \{a_k\}, \{0\}). \qquad (7.3) $$

The main result in this section is Theorem 7.2, which expresses the $\Lambda_k$ as a function of the sequences $a_k$ and $b_k$, thereby providing the concurrence (7.3).

To prove Theorem 7.2 we need to recall a classical result and notation. The Hankel determinant [6, Theorem 4.2] will be used:

$$ \det(\mu_{i+j}(\{b_k\}_{k \ge 0}, \{\lambda_k\}_{k \ge 0}))_{i,j=0}^n = \lambda_1^n \lambda_2^{n-1} \cdots \lambda_n^1. $$

Recall that for a sequence $a = \{a_k\}_{k \ge 0}$ we write $\delta a = \{a_{k+1}\}_{k \ge 0}$. We also define $\delta^{-1} a = \{a_{k-1}\}_{k \ge 0}$, where $a_{-1} = 1$ (the value of $a_{-1}$ is irrelevant for our purpose).

**Definition 7.1.** A Schröder path is a lattice path from $(r, 0)$ to $(s, 0)$, for some integers $r, s$, consisting of northeast steps $(1, 1)$, east steps $(1, 0)$, and south steps $(0, -1)$ that never goes below the $x$-axis. Given sequences $b = \{b_k\}_{k \ge 0}$ and $a = \{a_k\}_{k \ge 0}$, the weight $\mathrm{wt}(P)$ of a Schröder path $P$ is the product of $b_i$ for each east step starting at height $i$ and $a_i$ for each south step starting at height $i$.

Our main theorem of this section is the next theorem.

**Theorem 7.2.** Suppose that sequences $b = \{b_k\}_{k \ge 0}$, $a = \{a_k\}_{k \ge 0}$, and $\Lambda = \{\Lambda_k\}_{k \ge 0}$ satisfy

$$ \mu_{2n}(\{0\}, \{\Lambda_k\}) = \mu_n(\{b_k\}, \{a_k\}, \{0\}). $$

Then

$$ \Lambda_1 \Lambda_2 \cdots \Lambda_{2n} = \frac{f_n(a, b)}{f_{n-1}(a, b)}, $$

where

$$ f_n(a, b) = \sum_p \mathrm{wt}(p), $$

and the sum is over all $n$-tuples $p = (P_0, P_1, \dots, P_n)$ of non-intersecting Schröder paths, $P_k : (-k, 0) \to (k, 0)$, $0 \le k \le n$. Moreover,

$$ \Lambda_1 \Lambda_2 \cdots \Lambda_{2n-1} = a_0^{-1} \frac{f_n(\delta^{-1}a, \delta^{-1}b)}{f_{n-1}(\delta^{-1}a, \delta^{-1}b)}, $$

and if $a_k = b_k = 1$ then

$$ f_n(\{1\}, \{1\}) = 2^{\binom{n+1}{2}}. $$
*Proof.* Let

$$ \rho_n := \mu_{2n} (\{0\}, \{\Lambda_k\}) = \mu_n (\{b_k\}, \{a_k\}, \{0\}), $$

$$ \Delta_n := \det(\rho_{i+j})_{0 \le i,j \le n}. $$

Using the odd-even trick $B_n = \Lambda_{2n+1} + \Lambda_{2n}$ and $\Theta_n = \Lambda_{2n-1}\Lambda_{2n}$, we have

$$ \rho_n = \mu_{2n} (\{0\}, \{\Lambda_k\}) = \mu_n(\{B_k\}, \{\Theta_k\}). $$

Therefore

$$ \Delta_n = \det(\mu_{i+j}(\{B_k\}, \{\Theta_k\}))_{0 \le i,j \le n} = \Theta_1^n \Theta_2^{n-1} \cdots \Theta_n^1 = \Lambda_1^n \Lambda_2^n \Lambda_3^{n-1} \Lambda_4^{n-1} \cdots \Lambda_{2n-1}^1 \Lambda_{2n}^1, $$

which shows $\Lambda_1\Lambda_2\cdots\Lambda_{2n} = \Delta_n/\Delta_{n-1}$.

Kim and Stanton [18, Corollary 3.7] showed that $\mu_n (\{b_k\}, \{a_k\}, \{0\})$ is the sum of weights of all Schröder paths from $(0,0)$ to $(n,0)$. Since $\Delta_n = \det(\mu_{i+j} (\{b_k\}, \{a_k\}, \{0\}))_{0 \le i,j \le n}$, the $(n+1) \times (n+1)$ determinant $\Delta_n$ is the signed generating function for $(n+1)$-tuples of Schröder paths $(P_0, \dots, P_n)$, $P_k : (-k,0) \to (\sigma(k),0)$, for some permutation $\sigma$ of $\{0,1,\dots,n\}$. Because there are no SE edges ($\lambda_k = 0$), any two paths which intersect do so at integer coordinates. Thus we may apply the Lindström–Gessel–Viennot lemma of tail swapping to reduce this sum to non-intersecting paths, $\sigma = \text{identity}$, $P_k : (-k,0) \to (k,0)$. Thus $\Delta_n = f_n(a,b)$ and we obtain the identity for $\Lambda_1\Lambda_2\cdots\Lambda_{2n}$.

Now using the second odd-even trick $B'_n = \Lambda_{2n+2} + \Lambda_{2n+1}$ and $\Lambda'_n = \Lambda_{2n+1}\Lambda_{2n}$, we have

$$ \rho_{n+1} = \mu_{2n+2} (\{0\}, \{\Lambda_k\}) = \Lambda_1 \mu_n (\{B'_k\}, \{\Lambda'_k\}). $$

Then

$$ \begin{aligned} \Delta'_n &:= \det(\rho_{i+j+1})_{0 \le i,j \le n-1} = \Lambda_1^n \det(\mu_{i+j}(\{B'_k\}, \{\Lambda'_k\}))_{0 \le i,j \le n-1} \\ &= \Lambda_1^n \Lambda_2^{n-1} \Lambda_3^{n-1} \cdots \Lambda_{2n-2}^1 \Lambda_{2n-1}^1, \end{aligned} $$

so

$$ \Lambda_1 \Lambda_2 \cdots \Lambda_{2n-1} = \Delta'_n / \Delta'_{n-1}. $$

As in the even case, $\Delta'_n = \det(\mu_{i+j+1} (\{b_k\}, \{a_k\}, \{0\}))_{0 \le i,j \le n-1}$ is the generating function for $n$-tuples non-intersecting Schröder paths $p' = (P'_1, \dots, P'_n)$, $P'_k : (-k+1,0) \to (k,0)$. For $1 \le k \le n$, let $P_k$ be the path from $(-k,-1)$ to $(k,-1)$ obtained from $P'_k$ by adding a northeast step at the beginning and a south step at the end, and let $P_0$ be the empty path from $(0,-1)$ to $(0,-1)$. This gives a bijection from $n$-tuples non-intersecting Schröder paths $p' = (P'_1, \dots, P'_n)$, $P'_k : (-k+1,0) \to (k,0)$ to $(n+1)$-tuples non-intersecting Schröder paths $p = (P_0, P_1, \dots, P_n)$, $P_k : (-k,-1) \to (k,-1)$. Note that the starting point of $P_k$ has height $-1$, which shifts the indices of $a_k$ and $b_k$ down by one. This shows that

$$ \Delta'_n = a_0^{-n} \det(\mu_{i+j} (\{b_{k-1}\}, \{0\}, \{a_{k-1}\}))_{0 \le i,j \le n} = a_0^{-n} f_n(\delta^{-1}a, \delta^{-1}b), $$

and we obtain the identity for $\Lambda_1\Lambda_2\cdots\Lambda_{2n-1}$.

Finally, the fact that $\Delta_n = 2^{\binom{n+1}{2}}$ and $\Delta'_n = 2^{\binom{n+1}{2}}$ if $a_k = b_k = 1$ for all $k$ follows from [18, Theorem 6.15, A = B = 1, C = 0]. $\square$

The first few values of $\Lambda_1 \cdots \Lambda_k$ in Theorem 7.2 are
$$ \Lambda_1 = a_0^{-1} \frac{f_1(\delta^{-1}a, \delta^{-1}b)}{f_0(\delta^{-1}a, \delta^{-1}b)} = \frac{a_1 + b_0}{1}, $$

$$ \Lambda_1 \Lambda_2 = \frac{f_1(a, b)}{f_0(a, b)} = a_1 \frac{a_2 + b_1}{1}, $$

$$ \begin{aligned} \Lambda_1 \Lambda_2 \Lambda_3 &= a_0^{-1} \frac{f_2(\delta^{-1}a, \delta^{-1}b)}{f_1(\delta^{-1}a, \delta^{-1}b)} \\ &= a_1 \frac{a_1 a_2 a_3 + a_2^2 b_0 + a_2 a_3 b_0 + 2 a_2 b_0 b_1 + b_0 b_1^2 + a_1 a_2 b_2 + a_2 b_0 b_2}{a_1 + b_0}, \end{aligned} $$

$$ \begin{aligned} \Lambda_1 \Lambda_2 \Lambda_3 \Lambda_4 &= \frac{f_2(a, b)}{f_1(a, b)} \\ &= a_1 a_2 \frac{a_2 a_3 a_4 + a_3^2 b_1 + a_3 a_4 b_1 + 2 a_3 b_1 b_2 + b_1 b_2^2 + a_2 a_3 b_3 + a_3 b_1 b_3}{a_2 + b_1}. \end{aligned} $$

Remark 7.3. Eu and Fu [11] used the idea relating $\Delta_n$ and $\Delta'_{n-1}$ in the proof of Theorem 7.2 to give a simple proof of the Aztec diamond theorem, which is equivalent to the result $\Delta_n = 2^{\binom{n+1}{2}}$ when $a_k = b_k = 1$.

## 8. OPEN PROBLEMS

Recall that Kishore's theorem is a statement about the power series coefficients of the ratio $J_{\nu+1}(x)/J_{\nu}(x)$ of two Bessel functions.

**Theorem 8.1** (Kishore [19]). We have

$$ \frac{J_{\nu+1}(z)}{J_{\nu}(z)} = \sum_{n=1}^{\infty} \frac{N_{n,\nu}}{D_{n,\nu}} \left(\frac{z}{2}\right)^{2n-1}, $$

where

$$ D_{n,\nu} = \prod_{k=1}^{n} (k + \nu)^{\lfloor n/k \rfloor}, $$

and $N_{n,\nu}$ is a polynomial in $\nu$ with nonnegative integer coefficients.

We conjecture the following finite version of Kishore's theorem on a ratio of Lommel polynomials $L_{m,\nu}(x)$ defined in Section 3.

**Conjecture 8.2.** Let

$$ \frac{L_{m,\nu+2}(x)}{L_{m+1,\nu+1}(x)} = \sum_{n=0}^{\infty} \frac{N_{n,\nu}^{(m)}}{D_{n,\nu}^{(m)}} \left(\frac{x}{2}\right)^{2n+1}, $$

where

$$ D_{n,\nu}^{(m)} = \prod_{k=0}^{m} (\nu + k + 1)^{f(m,n,k)}, $$ 
$$ f(m,n,k) = \begin{cases} \max\left(\left\lfloor \frac{n+1}{k+1} \right\rfloor, \left\lfloor \frac{n+m-2k+1}{m-k+1} \right\rfloor\right), & \text{if } k \neq m/2, \\ 1, & \text{if } k = m/2. \end{cases} $$

Then $N_{n,\nu}^{(m)}$ is a polynomial in $\nu$ with nonnegative integer coefficients.
In Section 5 we saw that the ratio

$$ \frac{J_{\nu+1}^{(3)}(z; q^{-1})}{J_{\nu}^{(3)}(z; q^{-1})} = \frac{-q^{\nu+1}z}{1 - q^{\nu+1}} \cdot \frac{1\phi_1(0; q^{\nu+2}; q, q^{\nu+2}z^2)}{1\phi_1(0; q^{\nu+1}; q, q^{\nu+1}z^2)} $$

has two generalizations, the q-Nörlund continued fraction and Heine's continued fraction. These two generalizations seem to have a similar property as follows.

**Conjecture 8.3.** Let

$$ \sum_{n \ge 0} \gamma_n(a, b, c) z^n = \frac{2\phi_1(aq, bq; cq; q, z)}{2\phi_1(a, b; c; q, z)}. $$

Then

$$ \frac{\gamma_n(a, b, c)}{1-c} = \frac{P_n(a, b, c)}{\prod_{k=0}^{n} (1 - cq^k)^{\lfloor \frac{n+1}{k+1} \rfloor}}, $$

for some polynomial $P_n(a, b, c)$ in $a, b, c, q$ with integer coefficients.

**Conjecture 8.4.** Let

$$ \sum_{n \ge 0} \gamma'_n(a, b, c) z^n = \frac{2\phi_1(aq, b; cq; q, z)}{2\phi_1(a, b; c; q, z)}. $$

Then

$$ \frac{\gamma'_n(a, b, c)}{1-c} = \frac{P'_n(a, b, c)}{\prod_{k=0}^{n} (1 - cq^k)^{\lfloor \frac{n+1}{k+1} \rfloor}}, $$

for some polynomial $P'_n(a, b, c)$ in $a, b, c, q$ with integer coefficients.

Note that, between these two conjectures, only the second argument of the ${}_2\phi_1$ in the numerator differs (namely, $bq$ vs $b$).

**Problem 8.5.** Find a combinatorial proof of Theorem 5.12.

**Problem 8.6.** Find a combinatorial proof of Theorem 6.9, which contains the Bousquet-Mélou-Viennot result Theorem 6.8.

**Problem 8.7.** Find an Askey scheme whose top element is the associated Askey-Wilson polynomial which contains the q-Lommel polynomials. As an alternative, a referee has suggested that an Askey scheme with non-polynomial entries may exist which contains the q-Lommel polynomials.

**Acknowledgments.** The first author thanks the National Research Foundation for its support, via the grants NRF #2022R1A2C101100911 and NRF #2016R1A5A1008055.

## REFERENCES

[1] Elena Barcucci, Alberto Del Lungo, Jean-Marc Fédou, and Renzo Pinzani. Steep polyominoes, q-Motzkin numbers and q-Bessel functions. Discrete Mathematics, 189(1-3):21-42, 1998.

[2] Elena Barcucci, Alberto Del Lungo, Elisa Pergola, and Renzo Pinzani. Some permutations with forbidden subsequences and their inversion number. Discrete Mathematics, 234(1-3):1-15, 2001.

[3] Mireille Bousquet-Mélou and Jean-Marc Fédou. The generating function of convex polyominoes: the resolution of a q-differential system. Discrete Mathematics, 137(1-3):53-75, 1995.
[4] Mireille Bousquet-Mélou and Xavier Gérard Viennot. Empilements de segments et $q$-énumération de polyominos convexes dirigés. *Journal of Combinatorial Theory, Series A*, 60(2):196-224, 1992.

[5] Richard Brak and Anthony J. Guttmann. Exact solution of the staircase and row-convex polygon perimeter and area generating function. *Journal of Physics A.*, 23(20):4581-4588, 1990.

[6] Theodore S. Chihara. *An Introduction to Orthogonal Polynomials*. Gordon and Breach Science Publishers, 1978. Mathematics and its Applications, Vol. 13.

[7] Johann Cigler and Christian Krattenthaler. Bounded Dyck paths, bounded alternating sequences, orthogonal polynomials, and reciprocity. *European Journal of Combinatorics*, 2023. To appear.

[8] Annie Cuyt, Vigdis Brevik Petersen, Brigitte Verdonk, Haakon Waadeland, and William B. Jones. *Handbook of Continued Fractions for Special Functions*. Springer, 2008.

[9] Marie-Pierre Delest and Jean-Marc Fédou. Enumeration of skew Ferrers diagrams. *Discrete Mathematics*, 112(1-3):65-79, 1993.

[10] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.1.10 of 2023-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.

[11] Sen-Peng Eu and Tung-Shan Fu. A simple proof of the Aztec diamond theorem. *Electronic Journal of Combinatorics*, 12:Research Paper 18, 8, 2005.

[12] Jean-Marc Fédou. Combinatorial objects enumerated by $q$-Bessel functions. *Reports on Mathematical Physics*, 34(1):57-70, 1994.

[13] Philippe Flajolet. Combinatorial aspects of continued fractions. *Discrete Mathematics*, 32(2):125-161, 1980.

[14] George Gasper and Mizan Rahman. *Basic Hypergeometric Series*, volume 96 of *Encyclopedia of Mathematics and its Applications*. Cambridge University Press, second edition, 2004. With a foreword by Richard Askey.

[15] Mourad E. H. Ismail. *Classical and Quantum Orthogonal Polynomials in One Variable*, volume 98 of *Encyclopedia of Mathematics and its Applications*. Cambridge University Press, 2009.

[16] Mourad E. H. Ismail and David R. Masson. Generalized orthogonality and continued fractions. *Journal of Approximation Theory*, 83(1):1-40, 1995.

[17] Jang Soo Kim and Dennis Stanton. Ratios of Hahn-Exton $q$-Bessel functions and $q$-Lommel polynomials. *arXiv*, 2021.

[18] Jang Soo Kim and Dennis Stanton. Combinatorics of orthogonal polynomials of type $R_I$. *Ramanujan Journal*, 61(2):329-390, 2023.

[19] Nand Kishore. The Rayleigh polynomial. *Proceedings of the American Mathematical Society*, 15:911-917, 1964.

[20] David A. Klarner and Ronald L. Rivest. Asymptotic bounds for the number of convex $n$-ominoes. *Discrete Mathematics*, 8:31-40, 1974.

[21] Hendrik T. Koelink. Some basic Lommel polynomials. *Journal of Approximation Theory*, 96(2):345-365, 1999.

[22] Hendrik T. Koelink and René F. Swarttouw. On the zeros of the Hahn-Exton $q$-Bessel function and associated $q$-Lommel polynomials. *Journal of Mathematical Analysis and Applications*, 186(3):690-710, 1994.

[23] Hendrik T. Koelink and Walter Van Assche. Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton $q$-Bessel function. *Constructive Approximation*, 11(4):477-512, 1995.

[24] Christian Krattenthaler. Lattice path enumeration. In *Handbook of Enumerative Combinatorics*, pages 589-678. CRC Press, 2015.

[25] Jean-Claude Lalanne. $q$-Bessel, arbres, et chemins valués. *Journal of Combinatorial Theory. Series A*, 60(2):225-245, 1992.

[26] Jean-Claude Lalanne. Polyominos parallélogrammes à franges et fonctions de Bessel. *Discrete Mathematics*, 115(1-3):217-230, 1993.

[27] Derrick H. Lehmer. Zeros of the Bessel Function $J_n(x)$. *Mathematical Tables and Other Aids to Computation*, 1(10):405-407, 1945.

[28] Gérard Viennot. *Une Théorie combinatoire des polynômes orthogonaux généraux*. UQAM, 1983.
# ON *q*-ORDER STATISTICS

MALVINA VAMVAKARI¹

¹HAROKOPIO UNIVERSITY OF ATHENS, DEPARTMENT OF INFORMATICS AND TELEMATICS, ATHENS, GREECE; https://dblp.org/pid/46/6916.html

**Abstract.** Building on the notion of *q*-integral introduced by Thomae in 1869, we introduce *q*-order statistics (that, is *q*-analogues of the classical order statistics, for $0 < q < 1$) which arise from dependent and not identically distributed *q*-continuous random variables and to study their distributional properties. We study the *q*-distribution functions and the *q*-density functions of the relative *q*-ordered random variables. We focus on *q*-ordered variables arising from dependent and not identically *q*-uniformly distributed random variables and we derive their *q*-distributions, including *q*-power law, *q*-beta and *q*-Dirichlet distributions.

**Keywords:** *q*-order statistics, *q*-multinomial formulae, univariate and multivariate *q*-continuous random variables, *q*-uniform distribution, *q*-power law distribution, *q*-beta distribution, *q*-Dirichlet distribution, waiting times of the Heine process.

## 1. INTRODUCTION

Order statistics and their properties have been studied thoroughly the last decades. The literature devoted to order statistics from independent and identically distributed random variables is very extensive. The study of order statistics arising from independent or dependent and not identically distributed, random variables, is of great research interest. Excellent references devoted to order statistics are, among others, the work of Arnold, Balakrishnan and Nagaraja [2], Balakrishnan [3], David and Nagaraja [8], or Papadatos [13].

In the field of discrete *q*-distributions, Charalambides [6, p.167] has presented the order statistics arising from independent and identically distributed random variables, with common distribution a discrete *q*-uniform distribution. Charalambides [4, 5] also has studied the distributions of the record statistics in *q*-factorially increasing populations.

The main objective of this work is to introduce *q*-order statistics, for $0 < q < 1$, arising from dependent and not identically distributed *q*-continuous random variables and to study their distributional properties. We introduce *q*-order statistics as *q*-analogues of the classical order statistics. We study the *q*-distribution functions and *q*-density functions of the relative *q*-ordered random variables. We focus on *q*-ordered variables arising from dependent and not identically *q*-uniformly distributed random variables and we derive their *q*-distributions, including *q*-power law, *q*-beta and *q*-Dirichlet distributions. Moreover, we consider the Heine process, which had been introduced by Kyriakoussis and Vamvakari [12]; see also the work of Kemp [11]. Note that our notion of *q*-distribution is not related to the *q*-Gaussian distribution, or to other Tsallis distributions [14].
We prove that a conditional q-joint distribution of the waiting times of the Heine process coincides with the joint q-density function of q-ordered random variables arising from dependent q-continuous random variables.

This work contains three sections along with the introductory Section 1. In the preliminary Section 2, we present all our q-definitions. In the main Section 3, we state and prove our results concerning the q-order statistics and their distributional properties.

## 2. PRELIMINARIES, DEFINITIONS AND NOTATION

In this section, we define the q-series, the univariate and multivariate q-continuous random variables, the Heine process, and the q-uniform distribution. It will allow us to study q-order statistics in the next section.

### 2.1. q-Series preliminaries.
The q-shifted factorials are

$$ (a; q)_0 := 1, \quad (a; q)_n := \prod_{k=1}^{n} (1 - a q^{k-1}), \quad n = 1, 2, \dots, \text{ or } \infty. $$

The multiple q-shifted factorials are defined by

$$ (a_1, \dots, a_k; q)_n := \prod_{j=1}^{k} (a_j; q)_n $$

The q-binomial coefficient is defined by

$$ \begin{bmatrix} \nu \\ k \end{bmatrix}_q = \frac{(\nu; q)_n}{(\nu; q)_k (\nu; q)_{n-k}} = \frac{[\nu]_q!}{[k]_q! [n-k]_q}, \quad k = 0, 1, \dots, n, $$

where

$$ [n]_q! = [1]_q[2]_q \cdots [n]_q = \frac{(\nu; q)_n}{(1-q)^n} = \frac{\prod_{k=1}^{n} (1-q^k)}{(1-q)^n} $$

is the q-factorial number of order $n$ with $[t]_q = \frac{1-q^t}{1-q}$.

The $k$th-order factorial of the number $[n]_q$ is called $q$-factorial of $n$ of order $k$ and is given by

$$ [n]_k = [n]_q[n-1]_q \cdots [n-k+1]_q, \quad k = 1, 2, \dots, n. $$

Note that

$$ [n]_{q-1} = q^{-n+1}[n]_q, \quad [n]_{q-1}! = q^{-(n/2)}[n]_q! \quad \text{and} \quad \begin{bmatrix} n \\ k \end{bmatrix}_{\frac{1}{q}} = q^{-k(n-k)} \begin{bmatrix} n \\ k \end{bmatrix}_q. $$

The q-binomial coefficient $\begin{bmatrix} n \\ k \end{bmatrix}_q$, equals the $k$-combinations $\{m_1, \dots, m_k\}$ of the set $\{1, \dots, n\}$, weighted by $q^{m_1+\dots+m_k-(\frac{k+1}{2})}$,

$$ \sum_{1 \le m_1 < \dots < m_k \le n} q^{m_1 + \dots + m_k - \binom{k+1}{2}} = \begin{bmatrix} n \\ k \end{bmatrix}_q. \qquad (2.1) $$

Let $n$ be a positive integer and let $x, y$ and $q$ be real numbers, with $q \ne 1$. Then, a version of $q$-Vandermonde's formula is

$$ \begin{bmatrix} x + y \\ n \end{bmatrix}_q = \sum_{k=0}^{n} \begin{bmatrix} n \\ k \end{bmatrix}_q q^{k(y-n+k)} [x]_k [y]_{n-k}. $$
An interesting $q$-identity deduced by the above version of $q$-Vandermonde's formula is

$$ \sum_{k=0}^{n} (-1)^k \begin{bmatrix} n \\ k \end{bmatrix}_q q^{\binom{k+1}{2} - n(y+k)} \frac{[y]_q}{[y+k]_q} = \frac{1}{\begin{bmatrix} y+n \\ n \end{bmatrix}_q}. \quad (2.2) $$

Note that from the above equation we have the corresponding $q^{-1}$-identity

$$ \sum_{k=0}^{n} (-1)^k \begin{bmatrix} n \\ k \end{bmatrix}_q q^{\binom{k+1}{2} + ny} \frac{[y]_q}{[y+k]_q} = \frac{1}{\begin{bmatrix} y+n \\ n \end{bmatrix}_{\frac{1}{q}}}. \quad (2.3) $$

The $q$-binomial formula is

$$ \prod_{i=1}^{n} (1 + tq^{i-1}) = \sum_{k=0}^{n} q^{\binom{k}{2}} \begin{bmatrix} n \\ k \end{bmatrix}_q t^k. $$

The above $q$-binomial formula, by replacing $q$ by $q^{-1}$ and $t$ by $-t$, becomes

$$ \prod_{i=1}^{n} (1 - tq^{-(i-1)}) = \sum_{k=0}^{n} (-1)^k q^{-(\binom{k}{2})} \begin{bmatrix} n \\ k \end{bmatrix}_{\frac{1}{q}} t^k = \sum_{k=0}^{n} (-1)^k q^{-(\binom{k}{2}) - k(n-k)} \begin{bmatrix} n \\ k \end{bmatrix}_q t^k. \quad (2.4) $$

The $q$-multinomial coefficient is defined for nonnegative integers $n$ and $k_i$'s by

$$ \begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_q = \frac{[n]_q!}{[k_1]_q! \dots [k_r]_q! [n - k_1 - \dots - k_r]_q!} $$

We then have the two following equivalent expressions for the $q^{-1}$-multinomial coefficient

$$ \begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_{\frac{1}{q}} = q^{-\binom{n}{2} + \sum_{j=1}^{r+1} \binom{k_j}{2}} \begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_q \quad (2.5) \\ = q^{-\sum_{j=1}^{r} k_j(n-k_1-\dots-k_j)} \begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_q. $$

An ordered set partition of $A$ is a sequence $(A_1, \dots, A_m)$ of non-empty disjoint subsets of $A$, such that $A_1 \cup \dots \cup A_m = A$. Using the notation from Flajolet and Sedgewick's book *Analytic Combinatorics* [9], ordered set partitions are accordingly defined by the symbolic formula $\text{Seq}(\text{Set}_{\ge 1})$, and thus have the (exponential) generating function $1/(2 - \exp(t))$ of Fubini numbers $\{F_n\}_{n \ge 0} = \{1, 1, 3, 13, 75, 541, \dots\}$. E.g., there are 13 ordered set partitions of $\{1,2,3\}$.

Charalambides [7] showed that the $q$-multinomial coefficient $\begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_q$ equals the number of ordered partitions of the set $\{1, \dots, n\}$ into $r+1$ subsets, $(A_1, \dots, A_{r+1})$ of size $(k_1, \dots, k_{r+1})$, if one associates a specific $q$-weight to each subset. Writing $A_j = \{m_{j,1}, \dots, m_{j,k_j}\}$, this weight is $q^{m_{j,1}+\dots+m_{j,k_j}-(\binom{k_j+1}{2})}$, and one has

$$ \begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_q = \sum_{A_1, \dots, A_r} \prod_{j=1}^{r} q^{m_{j,1} + \dots + m_{j,k_j} - \binom{k_j+1}{2}}, \quad (2.6) $$

where the summation is over all the above mentioned ordered partitions of $\{1, \dots, n\}$.
Vamvakari [15] earlier proved the following alternative summation expansion of the q-multinomial coefficient

$$ \sum \prod_{j=1}^{r} q^{k_{j,1}+2k_{j,2}+\cdots+nk_{j,n}-(\begin{matrix} k_{j+1} \\ 2 \end{matrix})} = \begin{bmatrix} n \\ k_1, \dots, k_r \end{bmatrix}_q, $$

where the summation is over all $k_{j,i} = 0, 1$ such that $\sum_{i=1}^n k_{j,i} = k_j$ (for $j = 1, \dots, r$).

We shall also use the following q-difference operator (which we also call “q-derivative”)

$$ d_q f(x) := \frac{f(x) - f(qx)}{(1-q)x}. \qquad (2.7) $$

We refer to [1, Chapter 10.2] or [6] for a more thorough discussion of its properties. It is clear that it is a discrete analogue of the derivative; it satisfies e.g.

$$ d_q x^n = \frac{1-q^n}{1-q} x^{n-1} = [n]_q x^{n-1} $$

and $d_q(f(x) \cdot g(x)) = g(x)d_qf(x) + f(qx)d_qg(x)$. What is more, for differentiable functions, one has

$$ \lim_{q \to 1} d_q f(x) = f'(x). $$

Now, following [1, Chapter 10.1], we define the q-integral by

$$ \int_{0}^{a} f(x) d_q x := \sum_{n=0}^{\infty} [aq^n - aq^{n+1}] f(aq^n), \qquad (2.8) $$

$$ \int_{a}^{b} f(x) d_q x := \int_{0}^{b} f(x) d_q x - \int_{0}^{a} f(x) d_q x. $$

In this context, $d_q$ is sometimes called the Fermat measure, and should not be confused with the above q-derivative, even if they are, in some sense, related. The q-integral over $[0, \infty)$ uses the division points $\{q^n : -\infty < n < \infty\}$ and is

$$ \int_{0}^{\infty} f(x) d_q x := (1-q) \sum_{n=-\infty}^{\infty} q^n f(q^n). $$

## 2.2. The Heine process.
Kyriakoussis and Vamvakari [12] introduced the Heine process as a q-analogue of the Poisson process. The Heine process is defined as follows.

**Definition 2.1** (Heine Process). A continuous time process $\{X(t), t > 0\}$, where $X(t)$ expresses the number of arrivals in a time interval $(0, t]$, is called Heine process with parameters $0 < q < 1$ and $\lambda > 0$, if the following three assumptions hold

(a) The process starts at time 0 with $X(0) = 0$.

(b) In each time interval of length $\delta = (1-q)t$, one has 1 arrival with probability $p(t)$, and 0 arrival with probability $1-p(t)$, where

$$ p(t) := \frac{\lambda(1-q)t}{1 + \lambda(1-q)t}. $$

That is,

$$ P(X(t) - X(qt) = 1) = p(t) \quad \text{and} \quad P(X(t) - X(qt) = 0) = 1 - p(t). $$
This implies, for any $k \ge 1$:

$$P(X(q^{k-1}t) - X(q^k t) = 1) = \frac{\lambda(1-q)q^{k-1}t}{1 + \lambda(1-q)q^{k-1}t},$$

$$P(X(q^{k-1}t) - X(q^k t) = 0) = \frac{1}{1 + \lambda(1-q)q^{k-1}t}.$$

Also, the Heine process has the Heine distribution:

$$P(X(t) = k) = e_q(-\lambda t) \frac{q^{\binom{k}{2}} (\lambda t)^k}{[k]_q!},$$

for $k \in \mathbb{N}$, with $e_q(z) = \prod_{i=1}^{\infty} (1 - (1-q)zq^{i-1})^{-1}$, $|z| < 1/(1-q)$.

**2.3. Univariate and multivariate q-continuous random variables.** Kyriakoussis and Vamvakari [12] presented the following definition of q-continuous random variables. For clarity, let us begin by presenting this concept for one random variable.

**Definition 2.2 (q-continuous).** A random variable $X$ is called $q$-continuous (or “Fermat integrable”, as we integrate over the Fermat measure defined in (2.8)) if there exists a non-negative function $f_q(x)$ (for $x \ge 0$) such that

$$P(\alpha < X \le \beta) = \int_{\alpha}^{\beta} f_q(x) d_q x.$$

The function $f_q(x)$ is called *q-density function* of the random variable $X$.

Note that, in particular, one has

$$\int_{0}^{\infty} f_q(x) d_q x = 1.$$

For the corresponding distribution function

$$F(x) = P(X \le x),$$

we have by definition

$$P(\alpha < X \le \beta) = F(\beta) - F(\alpha),$$

and, for $x \ge 0$,

$$F(x) = \int_{0}^{x} f_{q}(t) d_{q} t.$$

Taking the *q*-derivative of the above relation we have

$$d_q F(x) = f_q(x)$$

and by the definition of the *q*-derivative we obtain

$$f_q(x) = \frac{F(x) - F(qx)}{(1-q)x} = \frac{P(qx < X \le x)}{(1-q)x}.$$

Let us now present the case of tuples.
**Definition 2.3** (multivariate q-continuous). A k-variate random variable $\mathcal{X} = (X_1, \dots, X_k)$ is called q-continuous (or “Fermat integrable”, as we integrate over the Fermat measure defined in (2.8)) if there exists a non-negative function $f_q(x_1, \dots, x_k)$ such that

$$P(\alpha_1 < X_1 \le \beta_1, \dots, \alpha_k < X_k \le \beta_k) = \int_{\alpha_k}^{\beta_k} \dots \int_{\alpha_1}^{\beta_1} f_q(x_1, \dots, x_k) d_q x_1 \dots d_q x_k.$$

The function $f_q(x_1, \dots, x_k)$ is called q-density function of the k-variate random variable $\mathcal{X} = (X_1, \dots, X_k)$ or joint q-density function of the random variables $X_1, \dots, X_k$.

In particular, we have

$$\int_{0}^{\infty} \dots \int_{0}^{\infty} f_q(x_1, \dots, x_k) d_q x_1 \dots d_q x_k = 1.$$

For the corresponding joint distribution function

$$F(x_1, \dots, x_k) = P(X_1 \le x_1, \dots, X_k \le x_k)$$

we have

$$F(x_1, \dots, x_k) = \int_0^{x_k} \dots \int_0^{x_1} f_q(t_1, \dots, t_k) d_q t_1 \dots d_q t_k. \quad (2.9)$$

Building on the notation (2.7), let us define the partial q-derivatives by

$$\frac{\partial F(x_1, \dots, x_k)}{\partial_q x_k \dots \partial_q x_1} = (d_q x_k) \dots (d_q x_1) F(x_1, \dots, x_k).$$

Then, taking the partial q-derivatives of the relation (2.9), we have

$$\frac{\partial F(x_1, \dots, x_k)}{\partial_q x_k \dots \partial_q x_1} = f_q(x_1, \dots, x_k), \quad x_i > 0, \quad i = 1, \dots, k$$

and by the definition of the partial q-derivative we obtain

$$f_q(x_1, \dots, x_k) = \frac{P(qx_1 < X_1 \le x_1, \dots, qx_k < X_k \le x_k)}{(1-q)x_1 \dots (1-q)x_k}. \quad (2.10)$$

The marginal q-density functions of the random variables $X$, $i = 1, \dots, k$, are given by

$$f_{X_i}(x_i) = \int_0^\infty \dots \int_0^\infty f_{\mathcal{X}}(x_1, \dots, x_k) d_q x_1 \dots d_q x_{i-1} d_q x_{i+1} \dots d_q x_k, \quad i = 1, \dots, k.$$

For the needs of this work, we also define the conditional q-density function. Let $(X, Y)$ be a bivariate q-continuous random variable, with q-density function $f_q(x, y) \ge 0$, $x, y > 0$ and $f_q(y) > 0$, $y > 0$ the marginal q-density function of $Y$. Then the function

$$f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}, \quad x > 0$$

is a q-density function because

$$f_{X|Y}(x|y) \ge 0, \quad x > 0$$

and

$$\int_0^\infty f_{X|Y}(x|y) d_q x = \frac{1}{f_Y(y)} \int_0^\infty f_{X,Y}(x,y) d_q x = \frac{f_Y(y)}{f_Y(y)} = 1.$$
Since

$$P(qx < X \leq x | qy < Y \leq y) = \frac{P(qx < X \leq x, qy < Y \leq y)}{P(qy < Y \leq y)}$$

we confirm that

$$f_{X|Y}(x|y) = \frac{P(qx < X \leq x | qy < Y \leq y)}{(1-q)x} = \frac{\frac{P(qx < X \leq x, qy < Y < y)}{(1-q)x(1-q)y}}{\frac{P(qy < Y \leq y)}{(1-q)y}} = \frac{f_{X,Y}(x,y)}{f_Y(y)} \quad (2.11)$$

and we give the following definition of conditional q-density function.

**Definition 2.4 (conditional q-density).** Let $(X, Y)$ be a bivariate q-continuous random variable. Let $f_{X,Y}(x, y)$ be its q-density function and $f_Y(y)$ the marginal q-density function of $Y$. If $f_Y(y) > 0$ for $y > 0$, the function

$$f_{X|Y}(x|y) := \frac{f_{X,Y}(x,y)}{f_Y(y)}$$

is called conditional q-density function of the random variable $X$ given that $qy < Y < y$.

Let $(X_1, \dots, X_k)$ be a q-continuous k-variate random variable, with joint q-density function $f(x_1, \dots, x_k) \geq 0$, $x_i > 0$, $i = 1, \dots, k$. The conditional q-density function of a q-continuous r-variate random variable $(X_1, \dots, X_r)$ given a q-continuous $(k-r)$-variate random variable $(X_{r+1}, X_{r+2}, \dots, X_k)$ is expressed as

$$h_{(X_1, \dots, X_r) | (X_{r+1}, X_{r+2}, \dots, X_k)}(x_1, \dots, x_r | x_{r+1}, \dots, x_k) = \frac{f(x_1, \dots, x_r)}{g(x_{r+1}, x_{r+2}, \dots, x_k)}, \quad (2.12)$$

where $g(x_{r+1}, x_{r+2}, \dots, x_k) > 0$ is the marginal q-density function of the $(k-r)$-variate random variable $(X_{r+1}, X_{r+2}, \dots, X_k)$.

**2.4. On the q-continuous q-uniform distribution.** For the needs of this work we give the definition of the q-uniform distribution and derive easily its main characteristics and properties. The q-uniform distribution is defined as follows.

**Definition 2.5 (q-uniform).** Let $X$ be a q-continuous random variable with q-density function

$$f_q(x) = \begin{cases} \frac{1}{\beta}, & 0 \leq x \leq \beta, \\ 0, & x < 0 \text{ or } x > \beta, \end{cases} \quad (2.13)$$

where $\beta > 0$. The distribution of the random variable $X$ is called q-uniform distribution with parameter $\beta$.

Note that by the function (2.13) and the definition of the q-integral,

$$\int_0^\beta f_q(x) d_q x = \sum_{n=0}^\infty \beta (q^n - q^{n+1}) f_q(\beta q^n) = 1,$$

as required by the definition of a q-density function.
**Proposition 2.6.** The r-th q-moments of the q-uniform distribution is given by

$$ \mu_r = E(X^r) = \frac{\beta^r}{[r+1]_q}. \qquad (2.14) $$

In particular its q-mean and q-variance are given respectively by

$$ \mu_q = E(X) = \frac{\beta}{[2]_q} \quad \text{and} \quad \sigma_q^2 = \frac{\beta^2 q}{(1+q+q^2)(1+2q+q^2)}. $$

*Proof.* Using the q-density function (2.13) and the definition of the q-integral, the r-th q-moment of the q-uniform distribution,

$$ \mu_r = E(X^r) = \int_0^\beta x^r f_q(x) d_q x, $$

is easily obtained in the form (2.14). The q-mean and q-variance of X follows. □

*Remark 2.7.* Let X be a q-continuous random variable obeying a q-uniform distribution with parameter $\beta$, then the linearly transformed q-continuous random variable $Y = \frac{X}{\beta}$ obeys the q-uniform distribution with parameter $\beta = 1$. Indeed

$$ F_Y(y) = P(Y \le y) = P(X \le \beta y) = F_X(\beta y) = \int_{\alpha}^{\beta y} f_q(x) d_q x = y, \quad 0 \le y \le 1. $$

So

$$ f_Y(y) = \begin{cases} 1, & 0 \le y \le 1, \\ 0, & y < 0 \text{ or } y > 1. \end{cases} $$

In the following proposition, we show that the linear transformation $Y = \frac{X}{\beta}$ can be generalized by considering the transformation $Y = F_X(X)$, where $F_X(x)$ is a distribution function of a q-continuous random variable X.

**Proposition 2.8.** Let X be a q-continuous random variable with probability function $F_X(x)$, $x \in R$. Then the distribution of the q-continuous random variable $Y = F_X(X)$ is the q-uniform distribution with parameter $\beta = 1$.

*Proof.* The distribution function of the q-continuous random variable Y is given, for $0 \le y \le 1$, by

$$ F_Y(y) = P(Y \le y) = P(F_X(X) \le y) = P(X \le F_X^{-1}(y)) = F_X(F_X^{-1}(y)) = y. $$

So

$$ f_Y(y) = \begin{cases} 1, & 0 \le y \le 1, \\ 0, & y < 0 \text{ or } y > 1 \end{cases} $$

and the proposition follows. □
### 3. MAIN RESULTS

**3.1. On the distributions of q-ordered random variables.** Let a $\nu$-variate q-continuous random variable $\mathcal{X} = (X_1, \dots, X_\nu)$ be defined in a sample space $\Omega$. Then for the values $x_1 = X_1(\omega), \dots, x_\nu = X_\nu(\omega), \omega \in \Omega$ there is a permutation $(i_1, \dots, i_\nu)$ of the $\nu$ indices $\{1, \dots, \nu\}$, such that $x_{i_1} \le \dots < x_{i_{\nu-1}} \le x_{i_\nu}$. The k-th ordered random variable is denoted by $X_{(k)}$ and defined by

$$X_{(k)}(\omega) = x_{(k)}, \omega \in \Omega,$$

where $x_{(k)} = x_{i_k}, k = 1, \dots, \nu$. In particular, for $k = 1$ this gives $X_{(1)} = \min\{X_1, \dots, X_\nu\}$ and, for $k = \nu$ this gives $X_{(\nu)} = \max\{X_1, \dots, X_\nu\}$. Generally, the following inequalities hold:

$$0 \le X_{(1)} \le \dots \le X_{(\nu)} \le \beta,$$

for a positive real number $\beta$.

We now introduce the following definition of q-ordered random variables.

**Definition 3.1 (q-ordered).** Let $\mathcal{Y} = (Y_1, \dots, Y_\nu)$ be a $\nu$-variate q-continuous random variable and $Y_{(k)}, 1 \le k \le \nu$, be the corresponding k-th ordered random variables. Assume that $Y_{(k)}, 1 \le k \le \nu$, satisfy the inequalities

$$0 \le Y_{(1)} < qY_{(2)} < Y_{(2)} < \dots < Y_{(\nu-1)} < qY_{(\nu)} < Y_{(\nu)} \le \beta, \quad (3.1)$$

for a positive real number $\beta$. Then, $Y_{(k)}$ (for any $k$ such that $1 \le k \le \nu$) is called the k-th q-ordered random variables.

Let $Y_{(k)}, 1 \le k \le \nu$, be the k-th q-ordered random variables, where the non-ordered q-continuous random variables $Y_1, \dots, Y_\nu$, are dependent and not identically distributed. The non-ordered, dependent and not identically distributed, random variables $Y_i, i = 1, \dots, \nu$, are randomly selected from the same sample space and the corresponding k-th q-ordered random variables, $Y_{(k)}, 1 \le k \le \nu$, satisfy inequalities (3.1). Each non-ordered random variable $Y_i$ is thus defined on the set

$$R_{Y_i} := [0, q^{(i-1)\beta}] = \cup_{j=i}^{\nu} R_j, \\ \text{where } R_j := (q^j\beta, q^{j-1}\beta] \text{ for } j = 1, \dots, \nu-1 \text{ and } R_{\nu} := [0, q^{\nu-1}\beta]. \quad (3.2)$$

In particular, one has

$$\cup_{j=1}^{\nu} R_j = [0, \beta] \text{ and } R_i \cap R_j = \emptyset \text{ for } i \ne j.$$

Moreover, we assume that the non-ordered random variables $Y_i$'s are not identically distributed according to their definitions sets but they are distributed with the same functional form. Furthermore, the stochastic dependencies satisfied by the non-ordered random variables $Y_i$'s are explicitly defined hereafter.
For any integer $r$ between 1 and $\nu$, let $\{i_1, \dots, i_r\}$ be an $r$-combination of $\{1, \dots, \nu\}$ satisfying $i_1 < \dots < i_r$, and let $\{i_{r+1}, i_{r+2}, \dots, i_\nu\}$ be its complementary combination (i.e., one has $\{i_1, \dots, i_r\} \cup \{i_{r+1}, i_{r+2}, \dots, i_\nu\} = \{1, \dots, \nu\}$) satisfying $i_{r+1} < i_{r+2} < \dots < i_\nu$. Then, we assume that the non-ordered random variables $Y_i$'s satisfy the following dependence relations for $y \in [0, \beta]$:

$$P(Y_{i_r} \le y | Y_{i_1} \le y, \dots, Y_{i_{r-1}} \le y) = P(Y_{i_r} \le q^{r-1}y), \quad (3.3)$$

$$P(Y_{i_r} \le y | Y_{i_1} > y, \dots, Y_{i_{r-1}} > y) = P(Y_{i_r} \le y), \quad (3.4)$$

and

$$\begin{align}
& P(Y_{i_m} \le y | Y_{i_1} \le y, \dots, Y_{i_r} \le y, Y_{i_{r+1}} > y, \dots, Y_{i_{m-1}} > y) \nonumber \\
& = P(Y_{i_m} \le y | Y_{i_1} \le y, Y_{i_r} \le y) \nonumber \\
& = P(Y_{i_m, q} \le q^{i_m - (m-r)} y), \quad m = r+1, r+2, \dots, \nu. \tag{3.5}
\end{align}$$

The $q$-distribution functions of the maximum, minimum, and $k$-th $q$-ordered random variables (respectively $Y_{(1)}$, $Y_{(\nu)}$, and $Y_{(k)}$) are derived in the following lemma.

**Lemma 3.2.** Let $Y_1, \dots, Y_\nu$ be dependent $q$-continuous random variables, where

(a) Each $Y_i$ is defined on the set $R_{Y_i}$ from Formula (3.2).

(b) Each $Y_i$ has a $q$-distribution function $F_{Y_i}(y) = P(Y_i \le y)$, for $y \in R_{Y_i}$, of the same functional form and satisfies the dependence relations (3.3), (3.4), (3.5).

Then, the $q$-distribution function of the maximum $q$-ordered random variable $Y_{(\nu)} = \max\{Y_1, \dots, Y_\nu\}$, where $Y_{(i)}$, $i = 1, \dots, \nu$, satisfy inequalities (3.1), is given for $y \in [0, \beta]$ by

$$F_{Y_{(\nu)}}(y) = \prod_{i=1}^{\nu} F_{Y_i}(q^{i-1}y). \quad (3.6)$$

Moreover, the $q$-distribution function of the minimum $q$-ordered random variable $Y_{(1)} = \min\{Y_1, \dots, Y_\nu\}$, where $Y_{(i)}$, $i = 1, \dots, \nu$, satisfy inequalities (3.1), is given by

$$F_{Y_{(1)}}(y) = 1 - \prod_{i=1}^{\nu} (1 - F_{Y_i}(y)). \quad (3.7)$$

Furthermore, the $q$-distribution function of $k$-th $q$-ordered random variable $Y_{(k)}$, $1 \le k \le \nu$, where $Y_{(i)}$, $i = 1, \dots, \nu$, satisfy inequalities (3.1), is given for $y \in [0, \beta]$ by

$$F_{Y_{(k)}}(y) = \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} \prod_{j=1}^{r} F_{Y_{i_j}}(q^{j-1}y) \prod_{m=r+1}^{\nu} (1 - F_{Y_{im}}(q^{i_m - (m-r)}y)), \quad (3.8)$$

where the inner summation is over all $r$-combinations $\{i_1, \dots, i_r\}$ of the set $\{1, \dots, \nu\}$.
Proof. Let $F_{Y_{(\nu)}}(y)$ be the $q$-distribution function of $Y_{(\nu)} = \max\{Y_1, \dots, Y_\nu\}$, then

$$
\begin{align*}
F_{Y_{(\nu)}}(y) &= P(Y_{(\nu)} \le y) = P(\max\{Y_1, \dots, Y_\nu\} \le y) \\
&= P(Y_1 \le y, Y_2 \le y, \dots, Y_\nu \le y) \\
&= P(Y_1 \le y)P(Y_2 \le y|Y_1 \le y) \cdots P(Y_\nu \le y|Y_1 \le y, \dots, Y_{\nu-1} \le y). \tag{3.9}
\end{align*}
$$

By assumptions (a) and (b), still for $y \in [0, \beta]$, the above equation (3.9) becomes

$$
F_{Y_{(\nu)}}(y) = \prod_{i=1}^{\nu} F_{Y_i}(q^{i-1}y).
$$

Let also $F_{Y_{(1)}}(y), y \in [0, \beta]$, be the $q$-distribution function of $Y_{(1)} = \min\{Y_1, \dots, Y_\nu\}$, then

$$
\begin{align*}
F_{Y_{(1)}}(y) &= P(Y_{(1)} \le y) = 1 - P(Y_{(1)} > y) = 1 - P(\min\{Y_1, \dots, Y_\nu\} > y) \\
&= 1 - P(Y_1 > y, Y_2 > y, \dots, Y_\nu > y) \\
&= 1 - P(Y_1 > y)P(Y_2 > y|Y_1 > y) \cdots P(Y_\nu > y|Y_1 > y, Y_2 > y, \dots, Y_{\nu-1} > y) \\
&= 1 - (1 - P(Y_1 < y))(1 - P(Y_2 < y|Y_1 < y) \cdots (1 - P(Y_\nu < y|Y_1 < y, \dots, Y_{\nu-1} < y)). \tag{3.10}
\end{align*}
$$

By assumptions (a) and (b), the above equation (3.10) becomes

$$
F_{Y_{(1)}}(y) = 1 - \prod_{i=1}^{\nu} (1 - F_{Y_i}(y)), y \in [0, \beta].
$$

Now, let $F_{Y_{(k)}}(y)$ be the $q$-distribution function of $Y_{(k)}$. Then, the event $Y_{(k)} \le y$ occurs if and only if at least $k$ random variables from $\{Y_1, Y_\nu\}$ take values in the set $[0, y]$ while the remaining ones $\nu - k$ take values in the set $(y, \beta]$. More precisely, consider an $r$-combination $\{i_1, \dots, i_r\}$ of $\{1, \dots, \nu, \}$, with $i_1 < \dots < i_r$, and its complementary combination $\{i_{r+1}, i_{r+2}, \dots, i_\nu\}$, with $i_{r+1} < i_{r+2} < \dots < i_\nu$. Then the $q$-distribution function of $Y_{(k)}$ is expressed as

$$
\begin{align*}
F_{Y_{(k)}}(y) &= P(Y_{(k)} \le y) \\
&= \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} P(Y_{i_1} \le y, \dots, Y_{i_r} \le y, Y_{i_{r+1}} > y, \dots, Y_{i_\nu} > y) \\
&= \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} P(Y_{i_1} \le y, \dots, Y_{i_r} \le y) P(Y_{i_{r+1}} > y, \dots, Y_{i_\nu} > y | Y_{i_1} \le y, \dots, Y_{i_r} \le y) \\
&= \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} P(Y_{i_1} \le y) \cdots P(Y_{i_r} \le y | Y_{i_1} \le y, \dots, Y_{i_{r-1}} \le y) \\
&\qquad \cdot P(Y_{i_{r+1}} > y | Y_{i_1} \le y, \dots, Y_{i_r} \le y) \cdots P(Y_{i_\nu} > y | Y_{i_1} \le y, \dots, Y_{i_r} \le y, Y_{i_{r+1}} > y, \dots, Y_{i_{\nu-1}} > y), \tag{3.11}
\end{align*}
$$

still with an inner summation over all $r$-combinations $\{i_1, \dots, i_r\}$ of the set $\{1, \dots, \nu\}$.
By assumptions (a) and (b), Equation (3.11) becomes

$$
F_{Y_{(k)}}(y) = \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} \prod_{j=1}^{r} F_{Y_{ij}}(q^{j-1}y) \prod_{m=r+1}^{\nu} \left(1 - F_{Y_{im}}\left(q^{im} - (m-r)y\right)\right),
$$

where the inner summation is over all $r$-combinations $\{i_1, \dots, i_r\}$ of the set $\{1, \dots, \nu\}$. $\square$
In the next theorem, we assume that the non ordered random variables $Y_i, i = 1, 2, \dots$ are dependent, q-uniformly distributed on the sets $[0, q^{i-1}t], t > 0, i = 1, \dots, \nu$, respectively, and we use the above lemma 3.2, to derive the q-distribution function and the q-density function of the corresponding maximum, minimum and k-th q-ordered random variables $Y_{(k)}, k = 1, \dots, \nu$.

**Theorem 3.3.** Let $Y_1, \dots, Y_\nu$ be dependent q-continuous random variables, q-uniformly distributed on the sets $[0, q^{i-1}t], t > 0, i = 1, \dots, \nu$, respectively. Assume that the random variables $Y_i, i = 1, \dots, \nu$, satisfy the dependence relations (3.3), (3.4), (3.5). Then, for $y \in [0, t]$, we have the following q-distribution functions and q-density functions:

* For the maximum q-ordered random variable $Y_{(\nu)} = \max\{Y_1, \dots, Y_\nu\}$ we have

$$F_{Y_{(\nu)}}(y) = \frac{y^{\nu}}{t^{\nu}}$$

and

$$f_{Y_{(\nu)}}(y) = [\nu]_q \frac{y^{\nu-1}}{t^{\nu}}. \qquad (3.12)$$

* For the q-distribution function and q-density function of the minimum q-ordered random variable $Y_{(1)} = \min\{Y_1, \dots, Y_\nu\}$ we have

$$F_{Y_{(1)}}(y) = 1 - \prod_{i=1}^{\nu} \left(1 - \frac{y}{q^{i-1}t}\right)$$

and

$$f_{Y_{(1)}}(y) = \frac{[\nu]_q}{q^{\nu-1}t} \prod_{i=1}^{\nu-1} \left(1 - \frac{y}{q^{i-1}t}\right). \qquad (3.13)$$

* For the q-distribution function and the q-density function of the k-th q-ordered random variable $Y_{(k)}$ we have

$$F_{Y_{(k)}}(y) = \sum_{r=k}^{\nu} \left[ \begin{matrix} \nu \\ r \end{matrix} \right]_{\frac{1}{q}} \frac{y^r}{t^r} \prod_{i=1}^{\nu-r} \left(1 - \frac{y}{q^{i-1}t}\right)$$

and

$$f_{Y_{(k)}}(y) = \frac{[\nu]_q! q^{\binom{\nu-k}{2}}}{[k-1]_q! [\nu-k]_q! q^{\binom{\nu}{2} - \binom{k}{2}}} \frac{y^{k-1}}{t^k} \prod_{j=1}^{\nu-k} \left(1 - \frac{y}{q^{i-1}t}\right). \qquad (3.14)$$

*Proof.* The theorem assumptions allow us to use Equation (3.6); the q-distribution function of $Y_{(\nu)}$ is thus

$$F_{Y_{(\nu)}}(y) = \prod_{i=1}^{\nu} F_{Y_i}(q^{i-1}y) = \frac{y q y}{t q t} \dots \frac{q^{\nu-1}}{q^{\nu-1}t} = \frac{y^{\nu}}{t^{\nu}}.$$
Taking the $q$-derivative of the above relation we have that the $q$-density function of $Y_{(\nu)}$ is straightforwardly given for $y \in [0, t]$ by

$$f_{Y_{(\nu)}}(y) = d_q F_{Y_{(\nu)}}(y) = [\nu]_q \frac{y^{\nu-1}}{t^{\nu}}.$$

Note that

$$\int_{0}^{t} f_{Y_{(\nu)}}(y) d_q y = \int_{0}^{t} [\nu]_{q} \frac{y^{\nu-1}}{t^{\nu}} d_q y = 1,$$

which is coherent with the fact we have here a $q$-density function.

Also, the $q$-distribution function of $Y_{(1)}$, by Equation (3.7) of the previous lemma 3.2, is straightforwardly given for $y \in [0, t]$ by

$$F_{Y_{(1)}}(y) = 1 - \prod_{i=1}^{\nu} (1 - F_{Y_i}(y)) = 1 - \prod_{i=1}^{\nu} \left(1 - \frac{y}{t}\right).$$

Taking the $q$-derivative of the above relation and using the $q$-binomial formula (2.4), we have that the $q$-density function of $Y_{(\nu)}$ is expressed as

$$\begin{aligned}
f_{Y_{(1)}}(y) &= d_q F_{Y_{(1)}}(y) = -\sum_{k=0}^{\nu} (-1)^k q^{-\binom{k}{2}} \begin{bmatrix} \nu \\ k \end{bmatrix}_{\frac{1}{q}} \frac{[k]_q y^{k-1}}{t^k} \\
&= \frac{[n]_q}{t} \sum_{k=0}^{\nu-1} (-1)^{k-1} q^{-\binom{k}{2}} q^{-k(\nu-k)} \begin{bmatrix} \nu-1 \\ k-1 \end{bmatrix}_q \frac{y^{k-1}}{t^{k-1}} \\
&= \frac{[n]_q}{q^{\nu-1} t} \sum_{k=0}^{\nu-1} (-1)^{k-1} q^{-\binom{k-1}{2}} q^{-(k-1)(\nu-k)} \begin{bmatrix} \nu-1 \\ k-1 \end{bmatrix}_q \frac{y^{k-1}}{t^{k-1}} \\
&= \frac{[n]_q}{q^{\nu-1} t} \sum_{k=0}^{\nu-1} (-1)^{k-1} q^{-\binom{k-1}{2}} \begin{bmatrix} \nu-1 \\ k-1 \end{bmatrix}_{\frac{1}{q}} \frac{y^{k-1}}{t^{k-1}} = \frac{[\nu]_q}{q^{\nu-1} t} \prod_{i=1}^{\nu-1} \left(1 - \frac{y}{q^{i-1}t}\right).
\end{aligned}$$

Note that using the $q$-binomial formula (2.4) and the $q$-identity (2.2), we obtain

$$\begin{aligned}
\int_{0}^{t} f_{Y_{(1)}}(y) d_q y &= \frac{[\nu]_q}{q^{\nu-1}} \sum_{j=0}^{\nu-1} (-1)^j q^{-\binom{j}{2}} \begin{bmatrix} \nu-1 \\ j \end{bmatrix}_{\frac{1}{q}} \int_{0}^{t} \frac{y^j}{t^{j+1}} d_q y \\
&= \frac{[\nu]_q}{q^{\nu-1}} \sum_{j=0}^{\nu-1} (-1)^j q^{-\binom{j}{2}} q^{-j(\nu-1-j)} \begin{bmatrix} \nu-1 \\ j \end{bmatrix}_q \frac{1}{[j+1]_q} \\
&= [\nu]_q \sum_{j=0}^{\nu-1} (-1)^j q^{\binom{j+1}{2}} q^{-(j+1)(\nu-1)} \begin{bmatrix} \nu-1 \\ j \end{bmatrix}_q \frac{1}{[j+1]_q} = [\nu]_q \frac{1}{\begin{bmatrix} \nu \\ \nu-1 \end{bmatrix}_q} = 1,
\end{aligned}$$

which is coherent with the fact we have here a $q$-density function.
Moreover, thanks to Equation (3.8), the q-distribution function of $Y_{(k)}$ is given by

$$ 
\begin{aligned}
F_{Y_{(k)}}(y) &= \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} \prod_{j=1}^{r} F_{Y_{i_j}} (q^{j-1}y) \prod_{m=r+1}^{\nu} (1 - F_{Y_{im}} (q^{i_m - (m-r-1)}y)) \\
&= \sum_{r=k}^{\nu} \sum_{1 \le i_1 < \dots < i_r \le \nu} \frac{y}{q^{i_1-1}t} \frac{qy}{q^{i_2-1}t} \dots \frac{q^{r-1}y}{q^{i_r-1}t} \left(1 - \frac{y}{t}\right) \left(1 - \frac{y}{qt}\right) \dots \left(1 - \frac{y}{q^{\nu-r-1}t}\right) \\
&= \sum_{r=k}^{\nu} \frac{y^r}{t^r} \prod_{i=1}^{\nu-r} \left(1 - \frac{y}{q^{i-1}t}\right) \sum_{1 \le i_1 < \dots < i_r \le \nu} q^{-i_1 - \dots - i_r + \binom{r+1}{2}},
\end{aligned}
$$

where the inner summation is over the r-combinations {$i_1, \dots, i_r$} of the set {$1, \dots, \nu$}.

Applying the formula of the q-binomial coefficient (2.1) in the above equation, we obtain

$$ F_{Y_{(k)}}(y) = \sum_{r=k}^{\nu} \left[ \frac{\nu}{r} \right]_{\frac{1}{q}} \frac{y^r}{t^r} \prod_{i=1}^{\nu-r} \left(1 - \frac{y}{q^{i-1}t}\right), y \in [0, t]. $$

Taking the q-derivative of the above q-distribution function, using suitably the q-binomial formula (2.4) and conducting all the needed algebraic manipulations, we have that the q-density function of $Y_{(k)}$ for $1 \le k \le \nu$ is expressed (for $y \in [0, t]$) as

$$ 
\begin{aligned}
f_{Y_{(k)}}(y) &= d_q F_{Y_{(k)}}(y) = \sum_{r=k}^{\nu} q^{-r(\nu-r)} \left[ \frac{\nu}{r} \right]_q [r]_q \frac{y^{r-1}}{t^r} \prod_{i=1}^{\nu-r} \left(1 - \frac{y}{q^{i-1}t}\right) \\
&\quad + \sum_{r=k}^{\nu} q^{-r(\nu-r)} \left[ \frac{\nu}{r} \right]_q \frac{q^r y^r}{t^r} \sum_{j=0}^{\nu-r} (-1)^j q^{-\binom{j}{2} - j(\nu-r-j)} \left[ \frac{\nu-r}{j} \right]_q [j]_q \frac{y^{j-1}}{t^j} \\
&= \frac{[\nu]_q}{t} \sum_{r=k}^{\nu} q^{-(\nu-r)} q^{-(r-1)(\nu-r)} \left[ \frac{\nu-1}{r-1} \right]_q \frac{y^{r-1}}{t^{r-1}} \prod_{i=1}^{\nu-r} \left(1 - \frac{y}{q^{i-1}t}\right) \\
&\quad - \frac{1}{t} \sum_{r=k}^{\nu} q^{-r(\nu-r)} \left[ \frac{\nu}{r} \right]_q \frac{q^r y^r}{t^r} [\nu-r]_q q^{-(\nu-r-1)} \\
&\quad \times \sum_{j=0}^{\nu-r} (-1)^{j-1} q^{-\binom{j-1}{2} - (j-1)(\nu-r-j)} \left[ \frac{\nu-r-1}{j-1} \right]_q \frac{y^{j-1}}{t^{j-1}} \\
&= \frac{[\nu]_q}{t} \sum_{r=k}^{\nu} q^{-(\nu-r)} q^{-(r-1)(\nu-r)} \left[ \frac{\nu-1}{r-1} \right]_q \frac{y^{r-1}}{t^{r-1}} \prod_{i=1}^{\nu-r} \left(1 - \frac{y}{q^{i-1}t}\right) \\
&\quad - \frac{[\nu]_q}{t} \sum_{r=k}^{\nu-1} q^{-(\nu-r-1)} q^{-r(\nu-1-r)} \left[ \frac{\nu-1}{r} \right]_q \frac{y^r}{t^r} \prod_{i=1}^{\nu-r-1} \left(1 - \frac{y}{q^{i-1}t}\right) \\
&= \frac{[\nu]_q}{t} q^{-(\nu-k)} q^{-(k-1)(\nu-k)} \left[ \frac{\nu-1}{k-1} \right]_q \frac{y^{k-1}}{t^{k-1}} \prod_{i=1}^{\nu-k} \left(1 - \frac{y}{q^{i-1}t}\right) \\
&= q^{-k(\nu-k)} \frac{[\nu]_q!}{[k-1]_q! [\nu-k]_q!} \frac{y^{k-1}}{t^k} \prod_{j=1}^{\nu-k} \left(1 - \frac{y}{q^{i-1}t}\right) \\
&= \frac{[\nu]_q! q^{\binom{\nu-k}{2}}}{[k-1]_q! [\nu-k]_q! q^{\binom{\nu-k}{2} - \binom{k}{2}}} \frac{y^{k-1}}{t^k} \prod_{j=1}^{\nu-k} \left(1 - \frac{y}{q^{i-1}t}\right).
\end{aligned}
$$
Note that using suitably the q-binomial formula (2.4), the q-identity (2.2) and carrying out all the needed algebraic manipulations, we obtain

$$
\begin{align*}
\int_0^t f_{Y_{(k)}}(y) dq y &= \frac{[\nu]_q! q^{\binom{\nu-k}{2}}}{[k-1]_q! [\nu-k]_q! q^{\binom{\nu}{2}-(k)} t} \int_0^t \frac{y^{k-1}}{t^{k-1}} \prod_{j=1}^{\nu-k} \left(1 - \frac{y}{q^{i-1}t}\right) dq y \\
&= q^{-k(\nu-k)} \frac{[\nu]_q!}{[k-1]_q! [\nu-k]_q!} \sum_{j=0}^{\nu-k} (-1)^j q^{-\binom{j}{2}} \left[\begin{smallmatrix} \nu-k \\ j \end{smallmatrix}\right]_{\frac{1}{q}} \int_0^t \frac{y^{k+j-1}}{t^{j+k}} dq y \\
&= q^{-k(\nu-k)} \frac{[\nu]_q!}{[k]_q! [\nu-k]_q!} \sum_{j=0}^{\nu-k} (-1)^j q^{-\binom{j}{2}} q^{-j(\nu-k-j)} \left[\begin{smallmatrix} \nu-k \\ j \end{smallmatrix}\right]_q \frac{[k]_q}{[k+j]_q} \\
&= \frac{[\nu]_q!}{[k]_q! [\nu-k]_q!} \sum_{j=0}^{\nu-k} (-1)^j q^{\binom{j+1}{2}-(\nu-k)(k+j)} \left[\begin{smallmatrix} \nu-k \\ j \end{smallmatrix}\right]_q \frac{[k]_q}{[k+j]_q} \\
&= \frac{[\nu]_q!}{[k]_q! [\nu-k]_q!} \frac{1}{\left[\begin{smallmatrix} \nu \\ \nu-k \end{smallmatrix}\right]_q} = 1,
\end{align*}
$$

which is coherent with the fact we have here a *q*-density function.

<figure><img src="image_106.png" /></figure>

Remark 3.4. The random variables $Y_{(1)}$ and $Y_{(\nu)}$ follow $q$-power law distributions (see Formulas (3.12) and (3.13)) while the random variables $Y_{(2)}, \dots, Y_{(\nu-1)}$ follow $q$-beta distributions (see Formula (3.14)).

In the following lemma, we consider the non-ordered $q$-continuous random variables, $Y_1, \dots, Y_\nu$, being dependent and not identically distributed, and we derive the joint $q$-distribution function of the $q$-ordered random variables, $Y_{(1)}$ and $Y_{(\nu)}$ that satisfy inequalities (3.1).

**Lemma 3.5.** Let $Y_1, \dots, Y_\nu$ be dependent $q$-continuous random variables, where

(a) Each $Y_i$ is defined on the set $R_{Y_i}$ from Formula (3.2).

(b) Each $Y_i$ has a $q$-distribution function $F_{Y_i}(y) = P(Y_i \le y)$, for $y \in R_{Y_i}$, of the same functional form and satisfy the dependence relations (3.3), (3.4), (3.5).

Then, the joint $q$-distribution function of the $q$-ordered random variables

$$
Y_{(1)} = \min\{Y_1, \dots, Y_\nu\} \quad \text{and} \quad Y_{(\nu)} = \max\{Y_1, \dots, Y_\nu\},
$$

is given by

$$
F_{Y_{(1)}, Y_{(\nu)}}(y, z) = \prod_{i=1}^{\nu} F_{Y_i}(q^{i-1}z) - \prod_{i=1}^{\nu} (F_{Y_i}(q^{i-1}z) - F_{Y_i}(y)) \quad (3.15)
$$

with $y < q^{\nu-1}z, \nu \ge 1, y, z \in [0, \beta]$.
Proof. Let $F_{Y_{(1)},Y_{(\nu)}}(y,z)$, $y < q^{\nu-1}z$, $\nu \ge 1$, $y, z \in [0, \beta]$, be the joint q-distribution function of the random variables $Y_{(1)}$ and $Y_{(\nu)}$. Using the expression

$$P(Y_{(1)} \le y, Y_{(\nu)} \le z) = P(Y_{(\nu)} \le z) - P(Y_{(1)} > y, Y_{(\nu)} \le z)$$

we then have

$$
\begin{align*}
F_{Y_{(1)},Y_{(\nu)}}(y,z) &= P(Y_{(1)} \le y, Y_{(\nu)} \le z) = P(Y_{(\nu)} \le z) - P(Y_{(1)} > y, Y_{(\nu)} \le z) \\
&= P(Y_1 \le z, Y_2 \le z, \dots, Y_\nu \le z) - P(y < Y_1 \le z, y < Y_2 \le z, \dots, y < Y_\nu \le z) \\
&= P(Y_1 \le z)P(Y_2 \le z|Y_1 \le z) \cdots P(Y_\nu \le z|Y_1 \le z, Y_2 \le z, \dots, Y_{\nu-1} \le z) \\
&\quad - P(y < Y_1 \le z)P(y < Y_2 \le z|y < Y_1 \le z) \cdots \\
&\quad \cdot P(y < Y_\nu \le z|y < Y_1 \le z, y < Y_2 \le z, \dots, y < Y_{\nu-1} \le z). \tag{3.16}
\end{align*}
$$

By assumptions (a) and (b), Equation (3.16) becomes (for $y, z \in [0, \beta]$ such that $y < q^{\nu-1}z$):

$$F_{Y_{(1)},Y_{(\nu)}}(y,z) = \prod_{i=1}^{\nu} F_{Y_i}(q^{i-1}z) - \prod_{i=1}^{\nu} (F_{Y_i}(q^{i-1}z) - F_{Y_i}(y)). \quad \square$$

In the next theorem, we assume that the non ordered random variables $Y_i$ are dependent and q-uniformly distributed on the sets $[0, q^{i-1}t]$ (for $t > 0$), and we use the above lemma 3.5, to derive the joint q-distribution function and the joint q-density function of the q-ordered random variables $Y_{(1)}$ and $Y_{(\nu)}$.

**Theorem 3.6.** Let $Y_1, \dots, Y_\nu$ be dependent q-continuous random variables, q-uniformly distributed on the sets $[0, q^{i-1}t]$, $t > 0$, $i = 1, \dots, \nu$, respectively. Assume that the random variables $Y_i$ satisfy the dependence relations (3.3), (3.4), (3.5). Then, the joint q-distribution function and the joint q-density function of the q-ordered random variables, $Y_{(1)} = \min\{Y_1, \dots, Y_\nu\}$ and $Y_{(\nu)} = \max\{Y_1, \dots, Y_\nu\}$ are given respectively by

$$F_{Y_{(1)},Y_{(\nu)}}(y,z) = \frac{z^{\nu}}{t^{\nu}} - \frac{z^{\nu}}{t^{\nu}} \prod_{i=1}^{\nu} \left(1 - \frac{y}{q^{i-1}z}\right)$$

and

$$f_{Y_{(1)},Y_{(\nu)}}(y,z) = q^{-\nu+1}[\nu]_q[\nu-1]_q \frac{z^{\nu-2}}{t^{\nu}} \prod_{i=1}^{\nu-2} \left(1 - \frac{y}{q^i z}\right)$$

with $y < q^{\nu-1}z, \nu \ge 1, y, z \in [0, t]$.

Proof. With the conditions of the theorem, by Equation (3.15), the q-distribution function of the random variables $Y_{(1)}$ and $Y_{(\nu)}$ becomes

$$
\begin{align*}
F_{Y_{(1)},Y_{(\nu)}}(y,z) &= \prod_{i=1}^{\nu} F_{Y_i}(q^{i-1}z) - \prod_{i=1}^{\nu} (F_{Y_i}(q^{i-1}z) - F_{Y_i}(y)) \\
&= \frac{z q z}{t q t} \cdots \frac{q^{\nu-1} z}{q^{\nu-1} t} - \left(\frac{z}{t} - \frac{y}{t}\right) \left(\frac{q z}{q t} - \frac{y}{q t}\right) \left(\frac{q^2 z}{q^2 t} - \frac{y}{q^2 t}\right) \cdots \left(\frac{q^{\nu-1} z}{q^{\nu-1} t} - \frac{y}{q^{\nu-1} t}\right) \\
&= \frac{z^{\nu}}{t^{\nu}} - \frac{z^{\nu}}{t^{\nu}} \prod_{i=1}^{\nu} \left(1 - \frac{y}{q^{i-1}z}\right).
\end{align*}
$$
Taking the partial $q$-derivatives of the above joint $q$-distribution function and using the $q$-binomial formula (2.4), we have that the joint $q$-density function of the random variables $Y_{(1)}$ and $Y_{(\nu)}$ is expressed as

$$
\begin{aligned}
f_{Y_{(1)}, Y_{(\nu)}}(y, z) &= \frac{\partial F_{Y_{(1)}, Y_{(\nu)}}(y, z)}{\partial q z \partial q y} = -\frac{1}{\partial q z \partial q y} \frac{z^{\nu}}{t^{\nu}} \prod_{i=1}^{\nu} \left(1 - \frac{y}{q^{i-1}z}\right) \\
&= -\frac{1}{\partial q z \partial q y} \frac{z^{\nu}}{t^{\nu}} \sum_{r=0}^{\nu} (-1)^r q^{-\binom{r}{2}} \left[\frac{\nu}{r}\right]_{\frac{1}{q}}^{\nu} \frac{y^r}{z^r} \\
&= -\frac{1}{t^{\nu}} \sum_{r=0}^{\nu} (-1)^r q^{-\binom{r}{2}} q^{-r(\nu-r)} \left[\frac{\nu}{r}\right]_q [r]_q [\nu-r]_q y^{r-1} z^{\nu-r-1} \\
&= \frac{z^{\nu-2}}{t^{\nu}} [\nu]_q [\nu-1]_q \sum_{r=0}^{\nu} (-1)^{r-1} q^{-\binom{r}{2}} q^{-r(\nu-r)} \left[\frac{\nu-2}{r-1}\right]_q^{\nu-2} \frac{y^{r-1}}{z^{r-1}} \\
&= \frac{z^{\nu-2}}{q^{\nu-1} t^{\nu}} [\nu]_q [\nu-1]_q \sum_{r=0}^{\nu} (-1)^{r-1} q^{-\binom{r-1}{2}} q^{-(r-1)(\nu-r-1)} \left[\frac{\nu-2}{r-1}\right]_q^{\nu-2} \frac{y^{r-1}}{(qz)^{r-1}} \\
&= \frac{z^{\nu-2}}{q^{\nu-1} t^{\nu}} [\nu]_q [\nu-1]_q \sum_{j=0}^{\nu} (-1)^j q^{-\binom{j}{2}} q^{-j(\nu-2-j)} \left[\frac{\nu-2}{j}\right]_q^{\nu-2} \frac{y^j}{(qz)^j} \\
&= q^{-\nu+1} [\nu]_q [\nu-1]_q \frac{z^{\nu-2}}{t^{\nu}} \prod_{i=1}^{\nu-2} \left(1 - \frac{y}{q^i z}\right).
\end{aligned}
$$

Note that using suitably the $q$-binomial formula (2.4), the $q^{-1}$-identity (2.3) and carrying out all the needed algebraic manipulations, we obtain

$$
\begin{aligned}
& \int_0^t \int_0^{q^{\nu-1}z} f_{Y_{(1)}, Y_{(\nu)}}(y, z) d_q y d_q z \\
&= \frac{q^{-\nu+1}}{t^{\nu}} [\nu]_q [\nu-1]_q \sum_{j=0}^{\nu} (-1)^j q^{-\binom{j}{2}} q^{-j(\nu-2-j)-j} \begin{bmatrix} \nu-2 \\ j \end{bmatrix}_q \int_0^t \int_0^{q^{\nu-1}z} y^j z^{\nu-2-j} d_q y d_q z \\
&= \frac{q^{-\nu+1}}{t^{\nu}} [\nu]_q [\nu-1]_q \sum_{j=0}^{\nu} (-1)^j q^{-\binom{j}{2}} q^{-j(\nu-2-j)-j} \begin{bmatrix} \nu-2 \\ j \end{bmatrix}_q \frac{q^{(\nu-1)(j+1)} t^{\nu}}{[j+1]_q [\nu]_q} \\
&= q^{-\nu+2} [\nu-1]_q \sum_{j=0}^{\nu} (-1)^j q^{\binom{j+1}{2}+\nu-2} \begin{bmatrix} \nu-2 \\ j \end{bmatrix}_q \frac{[1]_q}{[1+j]_q} \\
&= q^{-\nu+2} [\nu-1]_q \frac{1}{\begin{bmatrix} \nu-1 \\ \nu-2 \end{bmatrix}_{\frac{1}{q}}}
\end{aligned}
$$

which is coherent with the fact we have here a $q$-density function. $\square$

<figure><img src="image_107.png" /></figure>

In the following lemma, we consider the non-ordered $q$-continuous random variables, $Y_1, \dots, Y_\nu$, being dependent and not identically distributed, and we derive the joint $q$-distribution function of the $q$-ordered random variables, $Y_{(k)}$ and $Y_{(r)}$, $1 \le k < r \le \nu$.
**Lemma 3.7.** Let $Y_1, \dots, Y_\nu$ be dependent q-continuous random variables, where

(a) Each $Y_i$ is defined on the set $R_{Y_i}$ from Formula (3.2).

(b) Each $Y_i$ has a q-distribution function $F_{Y_i}(y) = P(Y_i \le y), y \in R_{Y_i}$ of the same functional form and satisfies the dependence relations (3.3), (3.4), (3.5).

Then, the joint q-distribution function of the q-ordered random variables $Y_{(k)}$ and $Y_{(r)}$ for $1 \le k < r \le \nu$, where $Y_{(i)}$, $i = 1, \dots, \nu$, satisfy inequalities (3.1), is given by

$$F_{Y_{(k)}, Y_{(r)}}(y, z) = \sum_{j=r}^{\nu} \sum_{s=k}^{j} \sum_{n_1=1}^{s} \prod_{n_1,...,n_r=s+1}^{j} (F_{Y_{i_{n_1}}}(q^{n_1-1}y) - F_{Y_{i_{n_r}}}(y)) \cdot \prod_{n_3=j+1}^{\nu} (1 - F_{Y_{i_{n_3}}}(q^{i_{n_3}-(n_3-j)}z)), y < q^{r-k}z, 1 \le k < r \le \nu, y, z \in [0, \beta], \quad (3.17)$$

where the inner summation is over all pairwise disjoint subsets $\{i_1, \dots, i_s\}$ and $\{i_{s+1}, \dots, i_j\}$ of the set $\{1, \dots, \nu\}$ with $1 \le i_1 < \dots < i_s \le \nu$ and $1 \le i_{s+1} < i_{s+2} < \dots < i_j \le \nu$.

*Proof.* Let $F_{Y_{(k)},Y_{(r)}}(y,z) = P(Y_{(k)} \le y, Y_{(r)} \le z)$, $y < q^{r-k}z, 1 \le k < r \le \nu, y, z \in [0, \beta]$, be the joint q-distribution function of the random variables $Y_{(k)}$ and $Y_{(r)}$ with $1 \le k < r \le \nu$. Then, the events $Y_{(k)} \le y$ and $Y_{(r)} \le z$ occur if and only if at least $k$ random variables in $\{Y_1, \dots, Y_\nu\}$ take values in the set $[0, y]$, while $r-k$ random other variables take values in the set $(y, z]$, and the remaining ones take values in the set $(z, \beta]$, $1 \le k < r \le \nu$. So, for $y < q^{r-k}z, 1 \le k < r \le \nu, y, z \in [0, \beta]$, we have

$$\begin{align*}
F_{Y_{(k)}, Y_{(r)}}(y, z) &= P(Y_{(k)} \le y, Y_{(r)} \le z) \\
&= \sum_{j=r}^{\nu} \sum_{s=k}^{j} \sum_{1 \le i_{s+1} < i_{s+2} < \dots < i_j \le \nu} P(\{Y_{i_j} \le y\}_{j=1,\dots,s}, \{y < Y_{i_j} \le z\}_{j=s+1,\dots,j}, \{Y_{i_j} > z\}_{j=j+1,\dots,\nu}) \\
&= \sum_{j=r}^{\nu} \sum_{s=k}^{j} \sum_{1 \le i_{s+1} < i_{s+2} < \dots < i_j \le \nu} P(Y_{i_1} \le y) P(Y_{i_2} \le y | Y_{i_1} \le y) \cdots P(Y_{i_s} \le y | Y_{i_{s-1}} \le y, Y_{i_{s-1}} \le y, \dots, Y_{i_1} \le y) \\
&\quad \cdot P(y < Y_{i_{s+1}} \le z | Y_{i_1} \le y, \dots, Y_{i_s} \le y) \\
&\quad \cdot P(y < Y_{i_{s+2}} \le z | Y_{i_1} \le y, \dots, Y_{i_s} \le y, y < Y_{i_{s+1}} \le z) \cdots \\
&\quad \cdot P(y < Y_{i_j} \le z | Y_{i_1} \le y, \dots, Y_{i_s} \le y, y < Y_{i_{s+1}} \le z, \dots, y < Y_{i_{j-1}} \le z) \\
&\quad \cdot P(Y_{i_{j+1}} > z | Y_{i_1} \le y, \dots, Y_{i_s} \le y, y < Y_{i_{s+1}} \le z, \dots, y < Y_{i_j} \le z) \cdots \\
&\quad \cdot P(Y_{i_\nu} > z | Y_{i_1} \le y, \dots, Y_{i_s} \le y, y < Y_{i_{s+1}} \le z, \dots, y < Y_{i_j} \le z, Y_{i_{j+1}} > z, \dots, Y_{i_{\nu-1}} > z), \tag{3.18}
\end{align*}$$

where the inner summation is over all pairwise disjoint subsets $\{i_1, \dots, i_s\}$ and $\{i_{s+1}, \dots, i_j\}$ of the set $\{1, \dots, \nu\}$ with $1 \le i_1 < \dots < i_s \le \nu$ and $1 \le i_{s+1} < i_{s+2} < \dots < i_j \le \nu$.
By assumptions (a) and (b), Equation (3.18) becomes (for $y < q^{r-k}z$, $1 \le k < r \le \nu$, and $y, z \in [0, \beta]$)

$$F_{Y_{(k)}, Y_{(r)}}(y, z) = \sum_{j=r}^{\nu} \sum_{s=k}^{j} \sum_{\substack{1 \le i_1 < \dots < i_s \le \nu \\ 1 \le i_{s+1} < \dots < i_j \le \nu}} \prod_{n_1=1}^{s} F_{Y_{i_1}}(q^{n_1-1}y) \\ \cdot \prod_{n_2=s+1}^{j} (F_{Y_{i_2}}(q^{n_2-s-1}z) - F_{Y_{i_2}}(y)) \prod_{n_3=j+1}^{\nu} (1 - F_{Y_{i_3}}(q^{i_3-(n_3-j)z})), $$

where the inner summation is over all pairwise disjoint subsets $\{i_1, \dots, i_s\}$ and $\{i_{s+1}, \dots, i_j\}$ of the set $\{1, \dots, \nu\}$ with $1 \le i_1 < \dots < i_s \le \nu$ and $1 \le i_{s+1} < i_{s+2} < \dots < i_j \le \nu$. $\square$

In the next theorem, we use the above lemma 3.7 to derive the joint q-distribution function and the joint q-density function of the ordered random variables.

**Theorem 3.8.** Let $Y_1, \dots, Y_\nu$ be dependent q-continuous random variables, q-uniformly distributed on the sets $[0, q^{i-1}t]$, $t > 0$, $i = 1, \dots, \nu$, respectively. Assume that the random variables $Y_i$, $i = 1, \dots, \nu$, satisfy the dependence relations (3.3), (3.4), (3.5). Then, the joint q-distribution function and the joint q-density function of the q-ordered random variables, $Y_{(k)}$ and $Y_{(r)}$, for $1 \le k < r \le \nu$, are given respectively by

$$F_{Y_{(k)}, Y_{(r)}}(y, z) = \sum_{j=r}^{\nu} \sum_{s=k}^{j} \left[ \begin{array}{c} \nu \\ s, j-s \end{array} \right] \frac{y^s z^{j-s}}{t^s t^{j-s}} \prod_{i=1}^{j-s} \left(1 - \frac{y}{q^{i-1}z}\right) \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right) \quad (3.19)$$

and

$$f_{Y_{(k)}, Y_{(r)}}(y, z) = \frac{q^{-r(\nu-r)} q^{-k(r-k)} [\nu]_q! y^{k-1}}{[k-1]_q! [r-k-1]_q! [\nu-r]_q! t^r} z^{r-k-1} \prod_{i=1}^{r-k-1} \left(1 - \frac{y}{q^i z}\right) \prod_{m=1}^{\nu-r} \left(1 - \frac{z}{q^{m-1}t}\right) \quad (3.20)$$

with $y < q^{r-k}z$, $1 \le k < r \le \nu$, $y, z \in [0, t]$.

*Proof.* By Equation (3.17) and the q-multinomial formulas (2.6) and (2.5), the joint q-distribution function of $Y_{(k)}$ and $Y_{(r)}$ satisfies

$$F_{Y_{(k)}, Y_{(r)}}(y, z) = \sum_{j=r}^{\nu} \sum_{s=k}^{j} \sum_{\substack{1 \le i_1 < \dots < i_s \le \nu \\ 1 \le m_1 < m_2 < \dots < m_{j-s} \le \nu}} q^{\binom{s+1}{2}} q^{\binom{j-s+1}{2}} q^{-i_1 - \dots - i_s} q^{-m_1 - \dots - m_{j-s}} \\ \cdot \frac{y^s z^{j-s}}{t^s t^{j-s}} \left(1 - \frac{y}{z}\right) \left(1 - \frac{y}{qz}\right) \cdots \left(1 - \frac{y}{q^{j-s-1}z}\right) \left(1 - \frac{z}{t}\right) \left(1 - \frac{z}{qt}\right) \cdots \left(1 - \frac{z}{q^{\nu-j-1}t}\right) \\ = \sum_{j=r}^{\nu} \sum_{s=k}^{j} \left[ \begin{array}{c} \nu \\ s, j-s \end{array} \right] \frac{y^s z^{j-s}}{t^s t^{j-s}} \prod_{i=1}^{j-s} \left(1 - \frac{y}{q^{i-1}z}\right) \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right), \quad (3.21)$$

where the inner summation of the first equality, is over all pairwise disjoint subsets $\{i_1, i_2, \dots, i_s\}$ and $\{m_1, m_2, \dots, m_{j-s}\}$ of the set $\{1, \dots, \nu\}$ with $1 \le i_1 < \dots < i_s \le \nu$ and $1 \le m_1 < m_2 < \dots < m_{j-s} \le \nu$.
The above joint q-distribution (3.21), of the random variables $Y_{(k)}$ and $Y_{(r)}$, for $1 \le k < r \le \nu$ and $y < q^{r-k}z$, $y, z \in [0, t]$ can be written as

$$F_{Y_{(k)}, Y_{(r)}}(y, z) = \sum_{j=r}^{\nu} \frac{q^{-j(\nu-j)}}{t^j} \left[\frac{\nu}{j}\right]_q^{\nu-j} \prod_{m=1}^{j} \left(1 - \frac{z}{q^{m-1}t}\right) \sum_{s=k}^{j} \left[\frac{j}{s}\right]_q \left(\frac{z}{q^s}\right)^{j-s} \prod_{i=1}^{j-s} \left(1 - \frac{y}{q^{i-1}z}\right).$$

Taking the partial q-derivative of the inner sum over $y$, using suitably the q-binomial formula (2.4) and carrying out all needed algebraic manipulations, we obtain

$$\frac{\partial_q}{\partial_q y} \sum_{s=k}^{j} q^{-s(j-s)} \left[\frac{j}{s}\right]_q z^{j-s} \prod_{i=1}^{j-s} \left(1 - \frac{y}{q^{i-1}z}\right) \\
= [j]_q \left( \sum_{s=k}^{j} q^{-s(j-s)} \left[\frac{j-1}{s-1}\right]_q y^{s-1} z^{j-s} \prod_{i=1}^{j-s} \left(1 - \frac{y}{q^{i-1}z}\right) \right) - [j]_q \left( \sum_{s=k}^{j} q^{-(s+1)(j-s-1)} \left[\frac{j-1}{s}\right]_q \right. \\
\left. \qquad y^s z^{j-s-1} \prod_{i=1}^{j-s-1} \left(1 - \frac{y}{q^{i-1}z}\right) q^{-k(j-k)} [j]_q \left[\frac{j-1}{k-1}\right]_q y^{k-1} z^{j-k} \prod_{i=1}^{j-k} \left(1 - \frac{y}{q^{i-1}z}\right) \right).$$

So,

$$\frac{\partial_q F_{Y_{(k)}, Y_{(r)}}(y, z)}{\partial_q y} \\
= \sum_{j=r}^{\nu} q^{-j(\nu-j)} \left[\frac{\nu}{j}\right]_q \frac{1}{t^j} \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right) q^{-k(j-k)} [j]_q \left[\frac{j-1}{k-1}\right]_q y^{k-1} z^{j-k} \prod_{i=1}^{j-k} \left(1 - \frac{y}{q^{i-1}z}\right) \\
= \sum_{j=r}^{\nu} q^{-j(\nu-j)} q^{-k(j-k)} \left[\frac{\nu}{j}\right]_q \left[\frac{j-1}{k-1}\right]_q [j]_q \frac{1}{t^j} y^{k-1} z^{j-k} \prod_{i=1}^{j-k} \left(1 - \frac{y}{q^{i-1}z}\right) \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right) \\
= \frac{[\nu]_q! y^{k-1}}{[k-1]_q! [\nu-k]_q!} \sum_{j=r}^{\nu} q^{-j(\nu-j)} q^{-k(j-k)} \left[\frac{\nu-k}{j-k}\right]_q \frac{z^{j-k}}{t^j} \prod_{i=1}^{j-k} \left(1 - \frac{y}{q^{i-1}z}\right) \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right).$$

In the last sum of this equation, taking the partial q-derivative over $z$, and using suitably q-binomial formula (2.4), we get

$$\frac{\partial_q}{\partial_q z} \sum_{j=r}^{\nu} q^{-j(\nu-j)} q^{-k(j-k)} \left[\frac{\nu-k}{j-k}\right]_q \frac{1}{t^j} z^{j-k} \prod_{i=1}^{j-k} \left(1 - \frac{y}{q^{i-1}z}\right) \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right) \\
= \sum_{j=r}^{\nu} q^{-j(\nu-j)} q^{-k(j-k)} \left[\frac{\nu-k}{j-k}\right]_q [j-k]_q \frac{1}{t^j} z^{j-k-1} \prod_{i=1}^{j-k-1} \left(1 - \frac{y}{q^i z}\right) \prod_{m=1}^{\nu-j} \left(1 - \frac{z}{q^{m-1}t}\right) \\
- \sum_{j=r}^{\nu} q^{-j(\nu-j)-k(j-k)-(\nu-j-1)+j-k} \left[\frac{\nu-k}{j-k}\right]_q [\nu-j]_q \frac{z^{j-k}}{t^{j+1}} \prod_{i=1}^{j-k} \left(1 - \frac{y}{q^i z}\right) \prod_{m=1}^{\nu-j-1} \left(1 - \frac{z}{q^{m-1}t}\right).$$
It follows that

$$
\begin{align*}
& \frac{\partial_q}{\partial_q z} \sum_{j=r}^{\nu} q^{-j(\nu-j)} q^{-k(j-k)} \left[ \begin{matrix} \nu-k \\ j-k \end{matrix} \right]_q \frac{1}{t^j} z^{j-k} \prod_{i=1}^{j-k} \left( 1 - \frac{y}{q^{i-1}z} \right) \prod_{m=1}^{\nu-j} \left( 1 - \frac{z}{q^{m-1}t} \right) \\
&= [\nu-k]_q \sum_{j=r}^{\nu} q^{-j(\nu-j)} q^{-k(j-k)} \left[ \begin{matrix} \nu-k-1 \\ j-k-1 \end{matrix} \right]_q \frac{1}{t^j} z^{j-k-1} \prod_{i=1}^{j-k-1} \left( 1 - \frac{y}{q^i z} \right) \prod_{m=1}^{\nu-j} \left( 1 - \frac{z}{q^{m-1}t} \right) \\
&\quad - [\nu-k]_q \sum_{j=r}^{\nu} q^{-j(\nu-j)-k(j-k)-(\nu-j-1)+j-k} \left[ \begin{matrix} \nu-k-1 \\ j-k \end{matrix} \right]_q \frac{z^{j-k}}{t^{j+1}} \prod_{i=1}^{j-k} \left( 1 - \frac{y}{q^i z} \right) \prod_{m=1}^{\nu-j-1} \left( 1 - \frac{z}{q^{m-1}t} \right) \\
&= q^{-r(\nu-r)} q^{-k(r-k)} [\nu-k]_q \left[ \begin{matrix} \nu-k-1 \\ r-k-1 \end{matrix} \right]_q \frac{1}{t^r} z^{r-k-1} \prod_{i=1}^{r-k-1} \left( 1 - \frac{y}{q^i z} \right) \prod_{m=r}^{\nu-j} \left( 1 - \frac{z}{q^{m-1}t} \right).
\end{align*}
$$

From this identity, we get that the joint q-density function given by

$$
\begin{align*}
f_{Y_{(k)}, Y_{(r)}}(y, z) &= \frac{\partial_q^2 F_{Y_{(k)}, Y_{(r)}}(y, z)}{\partial_q z \partial_q y} \\
&= \frac{q^{-r(\nu-r)} q^{-k(r-k)} [\nu]_q! \ y^{k-1}}{[k-1]_q! [r-k-1]_q! [\nu-r]_q! \ t^r} z^{r-k-1} \prod_{i=1}^{r-k-1} \left(1 - \frac{y}{q^i z}\right) \prod_{m=r}^{\nu-j} \left(1 - \frac{z}{q^{m-1} t}\right)
\end{align*}
$$

with $y < q^{r-k}z$, $y, z \in [0, t]$.

Note that using suitably the q-binomial formula (2.4), the $q^{-1}$ and $q$-identities (2.2), (2.3) and carrying out all the needed algebraic manipulations, we obtain

$$
\begin{align*}
& \int_0^t \int_0^{q^{r-k}z} f_{Y_{(k)}, Y_{(r)}}(y, z) d_q y \, d_q z \\
&= \frac{q^{-r(\nu-r)} q^{-k(r-k)} [\nu]_q! t^{-r}}{[k-1]_q! [r-k-1]_q! [\nu-r]_q!} \\
& \quad \times \int_0^t \left( \int_0^{q^{r-k}z} y^{k-1} \prod_{i=1}^{r-k-1} \left(1 - \frac{y}{q^i z}\right) d_q y \right) z^{r-k-1} \prod_{m=r}^{\nu-j} \left(1 - \frac{z}{q^{m-1} t}\right) d_q z \\
&= \frac{q^{-r(\nu-r)} q^{-k(r-k)} [\nu]_q! t^{-r}}{[k-1]_q! [r-k-1]_q! [\nu-r]_q! [k]_q} \\
& \quad \times \sum_{m=0}^{r-k-1} (-1)^m q^{\binom{m+1}{2} + (r-k-1)k} \left[ \begin{matrix} r-k-1 \\ m \end{matrix} \right]_q \frac{[k]_q}{[k+m]_q} \int_0^t z^{r-1} \prod_{m=r}^{\nu-j} \left(1 - \frac{z}{q^{m-1} t}\right) d_q z \\
&= \frac{q^{-k(r-k)} [\nu]_q!}{[k-1]_q! [r-k-1]_q! [\nu-r]_q!} \frac{1}{\left[ \begin{matrix} r-1 \\ r-k-1 \end{matrix} \right]_q} \frac{q^k}{[k]_q [r]_q} \sum_{i=0}^{\nu-r} (-1)^i q^{\binom{i+1}{2} - (i+r)(\nu-r)} \left[ \begin{matrix} \nu-r \\ i \end{matrix} \right]_q \frac{[r]_q}{[r+i]_q} \\
&= \frac{[\nu]_q!}{[k]_q! [r-k-1]_q! [\nu-r]_q!} \frac{[r-k-1]_q! [k]_q!}{[r]_q!} \frac{1}{\left[ \begin{matrix} \nu \\ \nu-r \end{matrix} \right]_q} = 1,
\end{align*}
$$

which is coherent with the fact we have here a *q*-density function. Note also that the joint
$q$-distribution function and $q$-density function of the random variables $Y_{(1)}$ and $Y_{(\nu)}$ are
given respectively by (3.19) and (3.20), for $k = 1$ and $r = \nu$. $\square$
Remark 3.9. The bivariate random variables $(Y_{(k)}, Y_{(r)})$ for $1 \le k < r \le \nu$ with joint $q$-density function (3.20), follow $q$-Dirichlet distributions.

In the following proposition, we consider the non-ordered $q$-continuous random variables, $Y_1, \dots, Y_\nu$, being dependent and not identically distributed and we derive the joint distribution function of the $q$-ordered random variables, $Y_{(1)}, \dots, Y_{(\nu)}$.

**Proposition 3.10.** Let $(Y_1, \dots, Y_\nu)$ be a $q$-continuous $\nu$-variate random vector with joint $q$-density function $f(y_1, \dots, y_\nu)$. Then the $q$-density function of the $q$-ordered random vector $\mathcal{Y} = (Y_{(1)}, \dots, Y_{(\nu)})$ is given by

$$f_{\mathcal{Y}}(y_{(1)}, \dots, y_{(\nu)}) = \sum_{0 < y_{(1)} < qy_{(2)} < y_{(2)} < qy_{(3)} < \dots < y_{(\nu-1)} < qy_{(\nu)} < y_{(\nu)} < \beta,} f_{Y_{i_1}|(Y_{i_2}, \dots, Y_{i_\nu})}(y_{(1)}|y_{(2)}, \dots, y_{(\nu)}), \quad (3.22)$$

where the summation is over all permutations $(i_1, \dots, i_\nu)$ of $\{1, \dots, \nu\}$.

*Proof.* The joint $q$-density function is

$$\begin{align*}
f_{\mathcal{Y}}(y_{(1)}, \dots, y_{(\nu)}) &= \frac{P(qy_{(1)} < Y_{(1)} \le y_{(1)}, \dots, qy_{(\nu)} < Y_{(\nu)} \le y_{(\nu)})}{(1-q)y_{(1)}(1-q)y_{(2)} \cdots (1-q)y_{(\nu)}} \\
&= (1-q)^{-\nu} \prod_{i=1}^{\nu} y_{(i)}^{-1} \sum P(qy_{(1)} < Y_{i_1} \le y_{(1)}, \dots, qy_{(\nu)} < Y_{i_\nu} \le y_{(\nu)}) \\
&= (1-q)^{-\nu} \prod_{i=1}^{\nu} y_{(i)}^{-1} \sum P(qy_{(\nu)} < Y_{i_\nu} \le y_{(\nu)}) P(qy_{(\nu-1)} < Y_{i_{\nu-1}} \le y_{(\nu-1)}|qy_{(\nu)} < Y_{i_\nu} \le y_{(\nu)}) \\
&\quad \cdots P(qy_{(\nu)} < Y_{i_1} \le y_{(1)}|qy_{(1)} < Y_{i_1} \le y_{(1)}, \dots, qy_{(\nu)} < Y_{i_\nu} \le y_{(\nu)}), \tag{3.23}
\end{align*}$$

where the summation is over all permutations $(i_1, \dots, i_\nu)$ of $\{1, \dots, \nu\}$.

Applying Definition 2.4 on the dependent $q$-density function and the relations (2.11), (2.12), to the above equation (3.23), we obtain 3.22. $\square$

Next, we assume that the non ordered random variables $Y_i, i = 1, \dots, \nu$ are dependent and $q$-uniformly distributed on the sets $[0, q^{i-1}t], t > 0, i = 1, \dots, \nu$, respectively, and the joint $q$-density function of the $q$-ordered random variables $Y_{(1)}, \dots, Y_{(\nu)}$, is obtained in the following corollary of Proposition 3.10.

**Corollary 3.11.** Let $Y_1, \dots, Y_\nu$ be dependent $q$-continuous random variables, $q$-uniformly distributed on the sets $[0, q^{i-1}t], t > 0, i = 1, \dots, \nu$, respectively. Assume that the random variables $Y_i, i = 1, \dots, \nu$, satisfy the dependence relations (3.3), (3.4), (3.5). Then the joint $q$-density function of the $\nu$-variate $q$-continuous random vector $\mathcal{Y} = (Y_{(1)}, \dots, Y_{(\nu)})$ with $Y_{(k)}, k = 1, \dots, \nu$, the $k$-th $q$-ordered random variables, is given by

$$f_{\mathcal{Y}}(y_1, \dots, y_\nu) = \frac{[\nu]_q!}{q^{\binom{\nu}{2}} t^\nu}, \quad 0 < y_1 < qy_2 < y_2 < qy_3 < \dots < y_{\nu-1} < qy_\nu < y_\nu < t. \quad (3.24)$$
Proof. Let $Y_1, \dots, Y_\nu$ be dependent $q$-continuous random variables, with each $Y_i$ (for $i = 1, \dots, \nu$) $q$-uniformly distributed on the set $[0, q^{i-1}t]$ (for some $t > 0$). Applying (3.22) of the previous proposition 3.10, the joint $q$-density function of the $\nu$-variate $q$-continuous random vector $\mathcal{Y} = (Y_{(1)}, \dots, Y_{(\nu)})$ (where each $Y_{(k)}$, for $k = 1, \dots, \nu$, is the $k$-th $q$-ordered random variable) is given (for $0 < y_1 < qy_2 < y_2 < qy_3 < \dots < y_{\nu-1} < qy_\nu < y_\nu < t$) by

$$f_{\mathcal{Y}}(y_1, \dots, y_\nu) = \sum \frac{1}{q^{i_\nu-1}t} \frac{1}{q^{i_{\nu-1}-1}t} \cdots \frac{1}{q^{i_1-1}t} \\ = \frac{1}{t^\nu} \sum \frac{1}{\prod_i q^{i_j-1}},$$

where the summation is over all permutations $(i_1, \dots, i_\nu)$ of $\{1, \dots, \nu\}$.

So,

$$f_{\mathcal{Y}}(y_1, \dots, y_\nu) = \frac{[\nu]_{q^{-1}t}!}{t^\nu} = \frac{[\nu]_q!}{q^{(\nu)_2}t^\nu}.$$

Note that

$$\int_0^t \int_0^{qy_\nu} \int_0^{qy_{\nu-1}} \cdots \int_0^{qy_3} \int_0^{qy_2} f_{\mathcal{Y}}(y_1, \dots, y_\nu) d_q y_1 d_q y_2 \cdots d_q y_{\nu-2} d_q y_{\nu-1} d_q y_\nu \\ = \int_0^t \int_0^{qy_\nu} \int_0^{qy_{\nu-1}} \cdots \int_0^{qy_3} \int_0^{qy_2} \frac{[\nu]_q!}{q^{(\nu)_2}t^\nu} d_q y_1 d_q y_2 \cdots d_q y_{\nu-2} d_q y_{\nu-1} d_q y_\nu = 1,$$

which confirms that Equation (3.24) is a joint $q$-density function. $\square$

**3.2. On a conditional joint $q$-distribution of the waiting times of the Heine process and $q$-order statistics.** Let $T_k$ be the waiting time of the $k$th arrival in the Heine process $\{X(t), t > 0\}$ with parameters $\lambda$ and $q$. Let us stop the process at $T_\nu$, for some integer $\nu \ge 1$. Now, we study the joint $q$-density function of the waiting times $T_1, \dots, T_\nu$. In the next theorem we prove that this conditional joint $q$-density function coincides with the joint $q$-density function of a $q$-ordered random sample of size $\nu$, from the $q$-continuous uniform distribution in the set $[0, q^{i-1}t]$, $i = 1, \dots, \nu$.

**Theorem 3.12.** Let $T_k$ be the waiting time of the $k$th arrival of the Heine process $\{X(t), t > 0\}$ with parameters $\lambda$ and $q$. Then the joint $q$-density function of the waiting times $T_1, \dots, T_\nu$, in which the first $\nu$ events occur given that $X(t) = \nu$, $0 < t_1 < \dots < t_\nu < t$ with $t_i \in (q^{\nu-i+1}t, q^{\nu-i}t]$, $i = 1, \dots, \nu-1$, is given by

$$f_q(t_1, \dots, t_\nu | X(t) = \nu) = \frac{[\nu]_q!}{q^{(\nu)_2}t^\nu},$$

that is the joint $q$-density function of a $q$-ordered random sample of size $\nu$, from the $q$-continuous uniform distribution in the set $[0, q^{i-1}t]$, $i = 1, \dots, \nu$.
*Proof.* By using the expression (2.10) and the three basic assumptions of Definition 2.1, the conditional joint q-density function of the Heine process satisfies the equation

$$
\begin{align*}
& f_q (t_1, \dots, t_\nu | X(t) = \nu) q^{\nu-1} (1-q) t q^{\nu-2} (1-q) t \dots q (1-q) t \\
&= P (q^\nu t < T_1 \le q^{\nu-1} t, \dots, q^2 t < T_\nu \le qt \mid X(t) = \nu) \\
&= P (X(q^\nu t) = 0) \left( \prod_{i=1}^{\nu-1} P (X(q^i (1-q)t) = 1) \right) \frac{P (X((1-q)t) = 0)}{P (X(t) = \nu)} \\
&= e_q (-\lambda q^\nu t) \frac{\lambda q^{\nu-1} (1-q)t}{1 + \lambda q^{\nu-1} (1-q)t} \frac{\lambda q^{\nu-2} (1-q)t}{1 + \lambda q^{\nu-2} (1-q)t} \dots \frac{\lambda q (1-q)t}{1 + \lambda q (1-q)t} \frac{1}{1 + \lambda (1-q)t} \\
&\quad \cdot \left( e_q (-\lambda t) \frac{q^{\binom{\nu}{2}} (\lambda t)^\nu}{[\nu]_q!} \right)^{-1} .
\end{align*}
$$

So,

$$
f_q (t_1, \dots, t_\nu | X(t) = \nu) = \frac{[\nu]_q!}{q^{\binom{\nu}{2}} t^\nu}.
$$

Therefore, by Corollary 3.11, this conditional joint q-density function coincides with
q-ordered density from the claim of the theorem. $\square$

4. CONCLUDING REMARKS

In this work we have introduced *q*-order statistics, for $0 < q < 1$, arising from dependent and not identically *q*-continuous random variables, as *q*-analogues of the classical order statistics. We have studied their main properties concerning the *q*-distribution functions and *q*-density functions of the relative *q*-ordered random variables. We have concentrated on the *q*-ordered variables arising from dependent and not identically *q*-uniformly distributed random variables. The derived *q*-distributions include *q*-power law, *q*-beta and *q*-Dirichlet distributions. The motivation for introducing *q*-order statistics was given by studying the properties of the waiting times of the Heine process.

As further study we propose the introduction of *q*-order statistics arising from dependent and not identically discrete *q*-distributed random variables. Last but not least, in link with lattice paths combinatorics, we intend to study the relations between *q*-order statistics and *q*-random walks in $\mathbb{Z}^d$, building on [10].

5. ACKNOWLEDGMENT

The author would like to thank the anonymous referees and the editors for their helpful and insightful comments and suggestions.
REFERENCES

[1] Georges Andrews, Richard Ashey, and Ranjan Roy. *Special Functions*. Cambridge University Press, 1999.

[2] Barry C. Arnold, Narayanaswamy Balakrishnan, and Haikady Navada Nagaraja. *A First Course in Order Statistics*. John Wiley & Sons, 1992.

[3] Narayanaswamy Balakrishnan. *Permanents, order statistics, outliers, and robustness*. *Revista Matemática Complutense*, 20:7-107, 2007.

[4] Charalambos A. Charalambides. Distributions of record statistics in a geometrically increasing population. *Journal of Statistical Planning and Inference*, 137:2214-2225, 2007.

[5] Charalambos A. Charalambides. Distributions of record statistics in a *q*-factorially increasing increasing population. *Communication in Statistics-Theory and Methods*, 38:2042-2055, 2009.

[6] Charalambos A. Charalambides. *Discrete q-Distributions*. John Wiley & Sons, 2016.

[7] Charalambos A. Charalambides. *q-Multinomial and negative q-multinomial distributions*. *Communication in Statistics – Theory and Methods*, 50:5873-5898, 2021.

[8] Herbert Aron David and Haikady Navada Nagaraja. *Order Statistics*. John Wiley & Sons, 3rd edition, 2003.

[9] Philippe Flajolet and Robert Sedgewick. *Analytic Combinatorics*. Cambridge University Press, 2009.

[10] Thomas Kamalakis and Malvina G. Vamvakari. *q-Random walks on Z<sup>d</sup>, d = 1, 2, 3*. *Methodology and Computing in Applied Probability*, 23:947-969, 2021.

[11] Adrienne W. Kemp. *Heine-Euler extensions of the Poisson distribution*. *Commun. Stat., Theory Methods*, 21(3):571-588, 1992.

[12] Andreas Kyriakoussis and Malvina G. Vamvakari. Heine process as a *q*-analog of the Poisson process – Waiting and interarrival times. *Communication in Statistics – Theory and Methods*, 46:4088-4102, 2017.

[13] Nickos Papadatos. Maximizing the expected range from dependent observations under mean-variance information. *Statistics*, 50:596-629, 2016.

[14] Constantino Tsallis. Possible generalization of Boltzmann-Gibbs statistics. *Journal of Statistical Physics*, 52:479-487, 1988.

[15] Malvina G. Vamvakari. On multivariate discrete *q*-distributions – A multivariate *q*-Cauchy’s formula. *Communication in Statistics – Theory and Methods*, 49:6080-6095, 2020.
# DIFFERENTIAL ALGEBRAIC GENERATING FUNCTIONS
OF WEIGHTED WALKS IN THE QUARTER PLANE

THOMAS DREYFUS¹

¹INSTITUT DE RECHERCHE MATHÉMATIQUE AVANCÉE, U.M.R. 7501 UNIVERSITÉ DE STRASBOURG
ET C.N.R.S. 7, RUE RENÉ DESCARTES 67084 STRASBOURG, FRANCE;
https://sites.google.com/site/thomasdreyfusmaths/

**Abstract.** In the present paper we study the nature of the trivariate generating functions of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies a nontrivial algebraic differential equation in one of its variables, if and only if it satisfies a nontrivial algebraic differential equation in each of its variables.

**Keywords:** Random walks in quarter plane, elliptic functions, transcendence.

## 1. INTRODUCTION

**Framework.** Consider a walk with small steps in the nonnegative quadrant $\mathbb{Z}_{\ge 0}^2 = \{0, 1, 2, \dots\}^2$ starting from $P_0 := (0, 0)$, that is a succession of points

$$P_0, P_1, \dots, P_k,$$

where each $P_n$ lies in the quarter plane, where the moves (or steps) $P_{n+1} - P_n$ belong to $\{0, \pm 1\}^2$, and the probability to move in the direction $P_{n+1} - P_n = (i, j)$ may be interpreted as some weight parameter $d_{i,j} \in [0, 1]$, with $\sum_{(i,j) \in \{0, \pm 1\}^2} d_{i,j} = 1$. The model of the walk (or model for short) is the data of the $d_{i,j}$ and the step set of the walk is the set of directions with nonzero weights, that is

$$S = \{(i, j) \in \{0, \pm 1\}^2 \mid d_{i,j} \neq 0\}.$$

If $d_{0,0} = 0$ and if the nonzero $d_{i,j}$ all have the same value, we say that the model is unweighted. The following picture provides an example of a walk in the nonnegative quadrant:

<figure><img src="image_108.png" /></figure>
Such objects are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures; see [4, 6] and references therein.

The weight of the walk is defined to be the product of the weights of its component steps. For any $(i, j) \in \mathbb{Z}_{\ge 0}^2$ and any $k \in \mathbb{Z}_{\ge 0}$, we let $q_{i,j,k}$ be the sum of the weights of all walks reaching the position $(i, j)$ from the initial position $(0, 0)$ after $k$ steps. We introduce the corresponding trivariate generating function

$$Q(x, y; t) := \sum_{i,j,k \ge 0} q_{i,j,k} x^i y^j t^k.$$

Being the generating function of probabilities, $Q(x, y; t)$ converges for all $(x, y, t) \in \mathbb{C}^3$ such that $|x|, |y| \le 1$ and $|t| < 1$. Note that in several papers, as in [4], it is not assumed that $\sum d_{i,j} = 1$. However, after a rescaling of the $t$ variable, we may always reduce to this case.

**Statement of the main result.** As we will see in the sequel, this paper takes part in a long history of articles that study the algebraic and differential relations satisfied by $Q(x, y; t)$. For any choice of a variable $\star$ among $x, y, t$, we say that $Q(x, y; t)$ is $\partial_{\star}$-algebraic if there exists $n \in \mathbb{Z}_{\ge 0}$, such that there exists a nonzero multivariate polynomial $P_{\star} \in \mathbb{C}(x, y, t)[X_0, \dots, X_n]$, such that

$$0 = P_{\star}(Q(x, y; t), \dots, \partial_{\star}^n Q(x, y; t)).$$

We stress that in the above definition, it is equivalent to require $0 \neq P_{\star} \in \mathbb{Q}[X_0, \dots, X_n]$; see Remark 3.1. Otherwise, we say that the series $Q(x, y; t)$ is $\partial_{\star}$-differentially transcendental.

Since the three variables $x, y$ and $t$ play a different role, we might expect the series to be of different nature with respect to the three derivatives. The main result of this paper, quite unexpected at first sight, shows that it is not the case. More precisely, using results of this paper and combining them to partial cases already known (see the discussion in the sequel), we complete the proof of the following main theorem.

**Theorem 1.1.** *The following facts are equivalent:*

* The series $Q(x, y; t)$ is $\partial_x$-algebraic;

* The series $Q(x, y; t)$ is $\partial_y$-algebraic;

* The series $Q(x, y; t)$ is $\partial_t$-algebraic.

Note that an algorithm is given in [16, Section 5] to decide whether the generating function is differentially algebraic in the $x$ variable or not, but this does not provide the differential equation when it exists.

**State of the art.** More generally, the question of studying whether $Q(x, y; t)$ satisfies algebraic (resp. linear differential, resp. algebraic differential) equations attracted the attention of many authors in the last decade. In the unweighted case, the problem was first addressed in the seminal paper [4] and solved using several methods, such as combinatorics, computer algebra, complex analysis, and more recently, difference Galois theory; see [2, 3, 7–9, 19–21]. We refer to the introduction of [12] for a history of the cited results, from which it follows that Theorem 1.1 is valid for the unweighted models.
The main difficulty in generalizing those results to weighted models is that, contrary to the unweighted framework, there are infinitely many weighted models. However, certain unweighted results are still valid in the weighted cases, while some others are proved by a case-by-case argument, and therefore cannot be generalized straightforwardly. So beyond the generalization, we believe that replacing case-by-case proofs by systematic arguments has its own interest since it shows that the unweighted version of Theorem 1.1 has not appeared by accident in a finite number of cases, and illustrates a general phenomenon.

In many situations, the equivalence between the $\partial_x$-algebraicity and the $\partial_y$-algebraicity can be straightforwardly deduced in this weighted context from the proof of [8, Proposition 3.10]. In [7, Theorem 2] it was proved that the $\partial_t$-algebraicity implies the $\partial_x$-algebraicity. So it remains to show the converse. In [2], the authors show that all $\partial_x$-differentially algebraic unweighted models have a decoupling function. They use this property to prove the $\partial_t$-algebraicity in that case. In [8], using difference Galois theory, the authors show that such unweighted models admit a telescoping relation. We refer to [16] for precise definitions of the two latter notions. In [11], it is proved that the $\partial_x$-algebraic weighted models also have a telescoping relation. Finally in [16] the equivalence between the existence of a telescoping relation and the existence of decoupling functions is shown. This implies that a $\partial_x$-algebraic series admits a certain decomposition into elliptic functions.

The main difficulty is that the existence of such decompositions is proved for fixed values of $t$, so nothing is known about the dependence in $t$ of the coefficients. For instance, the function $x\Gamma(t)$, seen as a function of $x$ is simple for all fixed value of $t$ (it is rational!) but it is differentially transcendental with respect to $t$, due to Hölder's result. We then have to make a careful study of the $t$-dependence of such elliptic relations, and use some results of $\partial_t$-algebraicity of the Weierstrass function in [2]. Finally, we are able to show that the $\partial_x$-algebraicity implies the $\partial_t$-algebraicity. The following diagram summarizes the various contributions toward the proof of Theorem 1.1.

<figure><img src="image_109.png" /></figure>

**Structure of the paper.** The paper is organized as follows. In Section 2 we provide some reminders of objects appearing in the study of models of walks in the quarter plane. More precisely, we will study well-known properties of the kernel curve and explain how the generating function may be continued. We will also explain why Theorem 1.1 is correct in some degenerate cases that we may withdraw. In Section 3 we prove technical results on differential algebraicity. Some intermediate results stay valid in the framework of algebraic functions and/or solution of linear differential equations, but to simplify the exposition, we chose to present this section in a unified framework, making some intermediate results suboptimal. Finally Section 4 is devoted to the proof of Theorem 1.1. We split our study in two cases depending on whether the so-called group of the walk is finite or not.
## 2. KERNEL OF THE WALK

**2.1. Functional equation.** The kernel of the walk is the polynomial defined by

$$K(x, y; t) := xy(1 - tS(x, y)),$$

where $S(x, y)$ denotes the jump polynomial

$$ 
\begin{aligned}
S(x, y) &= \sum_{(i,j) \in \{0, \pm 1\}^2} d_{i,j} x^i y^j \\
&= A_{-1}(x) \frac{1}{y} + A_0(x) + A_1(x)y \\
&= B_{-1}(y) \frac{1}{x} + B_0(y) + B_1(y)x,
\end{aligned}
$$

with $A_i(x) \in x^{-1}\mathbb{R}[x]$, $B_i(y) \in y^{-1}\mathbb{R}[y]$ (we recall that we consider weights $d_{i,j} \in [0, 1]$). The kernel plays an important role in the so-called kernel method and the techniques we are going to apply will vary depending on its algebraic properties, that have been studied in [14] (when $t = 1$), and in [10, 11] (when $t \in (0, 1)$). The starting point is the following fundamental functional equation.

**Lemma 2.1.** *The generating function $Q(x, y; t)$ satisfies the functional equation*

$$K(x, y; t)Q(x, y; t) = xy + K(x, 0; t)Q(x, 0; t) + K(0, y; t)Q(0, y; t) - K(0, 0; t)Q(0, 0; t).$$

*Proof.* As a walk is either empty, or a smaller walk to which one added a step (removing the cases leaving the quarter-plane), one has the following combinatorial functional equation

$$Q(x, y; t) = 1 + tS(x, y)Q(x, y; t) - t \frac{B_{-1}(y)}{x}Q(0, y; t) - t \frac{A_{-1}(x)}{y}Q(x, 0; t) + t \frac{d_{-1, -1}}{xy}Q(0, 0; t),$$

where the last summand removes the corresponding double counting. Multiplying by $xy$, we get Lemma 2.1. □

**2.2. Degenerate cases.** Like in [10], we will discard the following degenerate cases.

**Definition 2.2 (Degenerate model).** Let us fix $t \in (0, 1)$. A model of walk is called degenerate if one of the following holds:

* $K(x, y; t)$ factors in non-constant polynomials in $\mathbb{C}[x, y]$,

* $K(x, y; t)$ has $x$-degree (or $y$-degree) less than or equal to 1.

The notion of degeneracy thus seems to depend upon the parameter $t$. However, we will see in Proposition 2.3 below that the model is degenerate for a value of $t \in (0, 1)$ if and only if it is degenerate for all values of $t \in (0, 1)$. So, to lighten the terminology, we prefer not to stress this $t$-dependence and we say “degenerate” rather than “$t$-degenerate”.

In what follows we will sometimes represent a family of models of walks with arrows. For instance, the family of models represented by $\mathcal{A}$ or, equivalently, $\{\uparrow, \downarrow, \leftarrow, \rightarrow\}$ corresponds to models with $d_{1,0} = d_{0,-1} = d_{-1,1} = d_{-1,0} = 0$ and nothing is assumed on the other $d_{i,j}$. We stress the fact that the other $d_{i,j}$ (the weight of the arrows above) are allowed to be 0. In the following results, the behavior of the kernel curve *never* depends on $d_{0,0}$. This is the reason why, to reduce the amount of notations, we have decided not to associate an arrow to $d_{0,0}$.
The following proposition has been proved in [14, Lemma 2.3.2] for $t = 1$, in [10, Proposition 1.2] for $t$ is transcendental over $\mathbb{Q}(d_{i,j})$, and in [11, Proposition 1.3] for the other values of $t$ in $(0, 1)$.

**Proposition 2.3.** Let us fix $t \in (0, 1)$. A model of walk is degenerate if and only if at least one of the following holds:

(a) There exists $i \in \{-1, 1\}$ such that $d_{i,-1} = d_{i,0} = d_{i,1} = 0$. This corresponds to the following families of models

<figure><img src="image_110.png" /></figure>

(b) There exists $j \in \{-1, 1\}$ such that $d_{-1,j} = d_{0,j} = d_{1,j} = 0$. This corresponds to the following families of models

<figure><img src="image_111.png" /></figure>

(c) All the weights are 0 except maybe $\{d_{-1,-1}, d_{0,0}, d_{1,1}\}$ or $\{d_{-1,1}, d_{0,0}, d_{1,-1}\}$. This corresponds to the following families of models

<figure><img src="image_112.png" /></figure>

In virtue of the following lemma, Theorem 1.1 is valid for the degenerate models of walks. Therefore we will focus on models that are not degenerate.

**Lemma 2.4.** Assume that the model of walk is degenerate. Then $Q(x, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$ (and thus is differentially algebraic in its three variables).

*Proof.* We use Proposition 2.3. Consider the cases (a), (b), and first configuration of the case (c). In the unweighted case it is proved in [4, Section 1.2] that $Q(x, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$. The proof is the same in the weighted context but, to be self-contained, let us sketch the proof here. In the first configuration of case (a) the generating function is the same as the corresponding generating function of a model in the upper half-plane $\mathbb{Z} \times \mathbb{N}$. The latter is classically known to be algebraic over $\mathbb{C}(x, y, t)$, see for instance [5, Proposition 2]. In the second configuration of case (a), we have a unidimensional walk on the $y$-axis and such series is known to be rational, and therefore algebraic over $\mathbb{C}(x, y, t)$. The case (b) is similar. In the first configuration of case (c), we are considering a unidimensional walk on the half-line $\{(x, x), x \in \mathbb{N}\}$, and the generating function is algebraic. Since in all these cases, $Q(x, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$, it is differentially algebraic in its three variables. In the last configuration of case (c), all the weights are 0 except maybe $\{d_{-1,1}, d_{0,0}, d_{1,-1}\}$, so the walk cannot leave $(0, 0)$ and we have

$$Q(x, y; t) = \sum_{k=0}^{\infty} d_{0,0}^k t^k = \frac{1}{1 - d_{0,0}t}.$$

Therefore the result holds in that case too.

□
2.3. **Genus of the walk.** The kernel curve $E_t$ is the complex affine algebraic curve defined by

$$E_t = \{(x, y) \in \mathbb{C} \times \mathbb{C} \mid K(x, y; t) = 0\}.$$

We shall now consider a compactification of this curve. We let $\mathbb{P}^1(\mathbb{C})$ be the complex projective line, that is the quotient of $(\mathbb{C} \times \mathbb{C}) \setminus \{(0,0)\}$ by the equivalence relation $\sim$ defined by

$$(x_0, x_1) \sim (x'_0, x'_1) \Leftrightarrow \exists \lambda \in \mathbb{C}^*, (x'_0, x'_1) = \lambda(x_0, x_1).$$

The equivalence class of $(x_0, x_1) \in (\mathbb{C} \times \mathbb{C}) \setminus \{(0,0)\}$ is denoted by $[x_0 : x_1] \in \mathbb{P}^1(\mathbb{C})$. The map $x \mapsto [x : 1]$ embeds $\mathbb{C}$ inside $\mathbb{P}^1(\mathbb{C})$. The latter map is not surjective: its image is $\mathbb{P}^1(\mathbb{C}) \setminus \{[1 : 0]\}$; the missing point $[1 : 0]$ is usually denoted by $\infty$. Now, we embed $E_t$ inside $\mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^1(\mathbb{C})$ via $(x, y) \mapsto ([x : 1], [y : 1])$. The kernel curve $\bar{E}_t$ is the closure of this embedding of $E_t$. In other words, the kernel curve $\bar{E}_t$ is the algebraic curve defined by

$$\bar{E}_t = \{([x_0 : x_1], [y_0 : y_1]) \in \mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^1(\mathbb{C}) \mid \bar{K}(x_0, x_1, y_0, y_1; t) = 0\}$$

where $\bar{K}(x_0, x_1, y_0, y_1; t)$ is the following degree two homogeneous polynomial in the four variables $x_0, x_1, y_0, y_1$

$$\bar{K}(x_0, x_1, y_0, y_1; t) = x_1^2 y_1^2 K \left( \frac{x_0}{x_1}, \frac{y_0}{y_1}; t \right) = x_0 x_1 y_0 y_1 - t \sum_{i,j=0}^{2} d_{i-1,j-1} x_0^i x_1^{2-i} y_0^j y_1^{2-j}.$$

Although it may seem more natural to take the closure of $\bar{E}_t$ in $\mathbb{P}^2(\mathbb{C})$, the above definition allows us to avoid unnecessary singularities.

The following proposition has been proved in [10, Proposition 2.1 and Corollary 2.6], when $t$ is transcendental over $\mathbb{Q}(d_{i,j})$ and has been extended for a general $0 < t < 1$ in [11, Proposition 1.9].

**Proposition 2.5.** Let us fix $t \in (0,1)$ and assume that the model of the walk is not degenerate. The following facts are equivalent:

(1) $\bar{E}_t$ is an elliptic curve;

(2) The set of authorized directions $S$ is not included in any half-space with boundary passing through the origin.

Let us now discuss the case where for $t \in (0,1)$ fixed, the model is not degenerate and $\bar{E}_t$ is not an elliptic curve. By Proposition 2.3 and Proposition 2.5, this corresponds to nondegenerate models that belong to one of the four families in Figure 1.

<figure><img src="image_113.png" /><figcaption>FIGURE 1. Our four nondegenerate models</figcaption></figure>

Note that although the third configuration in Figure 1 is called nondegenerate, it leads to walks that never escape from $(0,0)$ and thus their generating function is trivial.

The following lemma yields that Theorem 1.1 is valid for the families of models in Figure 1.
**Lemma 2.6.** The following holds:

(a) Assume that the model of the walk is not degenerate and belongs to the first family in Figure 1. Then $Q(x, y; t)$ is differentially transcendental in its three variables.

(b) Assume that the model of the walk belongs to the second, third or the fourth family in Figure 1. Then $Q(x, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$, and thus is differentially algebraic in its three variables.

*Proof.* (a) This is [7, Corollary 2.2]; see also [9, Theorem 3.1].

(b) Consider the second family. We have $Q(x, 0; t) = Q(0, 0; t)$ and $K(x, 0; t) = K(0, 0; t)$. Then by Lemma 2.1,

$$K(x, y; t)Q(x, y; t) = K(0, y; t)Q(0, y; t) + xy. \qquad (2.1)$$

Let us see that with the same arguments as for the walks in the half-plane, we deduce that $Q(x, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$. The idea is to locally write $K(\phi(y; t), y; t) = 0$. Evaluating at $(\phi(y; t), y; t)$ we then have for convenient $y$ and $t$, $0 = K(0, y; t)Q(0, y; t) + \phi(y; t)y$, proving that $Q(0, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$. The functional equation (2.1) allows then to conclude that $Q(x, y; t)$ is algebraic over $\mathbb{C}(x, y, t)$. As in the proof of Lemma 2.4, we may deduce that $Q(x, y; t)$ is differentially algebraic in its three variables. The reasoning for the fourth family is similar. For the third family, the walk has to stay at $(0, 0)$ and we have

$$Q(x, y; t) = \sum_{k=0}^{\infty} d_{0,0}^{k} t^{k} = \frac{1}{1 - d_{0,0}t}.$$

Therefore the result holds in that case too.

**2.4. Group of the walk.** From now on, we may focus on the case where $\bar{E}_t$ is an elliptic curve. Recall that we have seen in Proposition 2.3, that $K(x, y; t)$ has degree two in $x$ and $y$, and nonzero coefficient of degree 0 in $x$ and $y$. Hence, $A_1(x), A_{-1}(x), B_1(y), B_{-1}(y)$ are not identically zero.

Following [4, Section 3], [17, Section 3] or [14], and using the notations introduced in Section 2.3, we consider the rational involutions given by

$$i_1([x_0 : x_1], [y_0 : y_1]) = \left( \frac{x_0}{x_1}, \frac{A_1(\frac{x_0}{x_1})}{A_{-1}(\frac{x_0}{x_1}) \frac{y_0}{y_1}} \right) \quad \text{and} \quad i_2([x_0 : x_1], [y_0 : y_1]) = \left( \frac{B_{-1}(\frac{y_0}{y_1})}{B_1(\frac{y_0}{y_1}) \frac{x_0}{x_1}}, \frac{y_0}{y_1} \right).$$

Note that we have $i_1([x_0/x_1 : 1], [y_0/y_1 : 1]) = i_1([x_0 : x_1], [y_0 : y_1])$ and the same holds for $i_2$. Note also that $i_1, i_2$ are a priori not defined when the denominators vanish but we will see in the sequel that we may overcome this problem when we will restrict to $\bar{E}_t$.

For a fixed value of $x$, there are at most two possible values of $y$ such that $(x, y) \in \bar{E}_t$. The involution $i_1$ corresponds to interchanging these values. A similar interpretation can be given for $i_2$. Therefore the kernel curve $\bar{E}_t$ is left invariant by the natural action of $i_1, i_2$.
<figure><img src="image_114.png" /><figcaption>FIGURE 2. The maps $i_1$ and $i_2$ restricted to the kernel curve $\bar{E}_t$ are denoted by $\iota_1$ and $\iota_2$, respectively.</figcaption></figure>

We denote by $\iota_1, \iota_2$ the restriction of $i_1, i_2$ to $\bar{E}_t$; see Figure 2. Again, these functions are a priori not defined where the denominators vanish. However, by [10, Proposition 4.1], this is only an “apparent problem”. To be precise, the authors proved this for $t$ transcendental over $\mathbb{Q}(d_{i,j})$ but the proof is still valid when $\bar{E}_t$ is an elliptic curve. We then obtain that $\iota_1$ and $\iota_2$ can be extended to morphisms of $\bar{E}_t$. We recall that a rational map $f$ from $\bar{E}_t$ to $\bar{E}_t$ is a morphism if it is regular at any $P \in \bar{E}_t$, i.e. if $f$ can be represented in suitable affine charts containing $P$ and $f(P)$ by a rational function with nonvanishing denominator at $P$.

Let us finally define $\sigma = \iota_2 \circ \iota_1$. Note that such a map is called a QRT-map and has been widely studied; see [13].

**Definition 2.7 (Group of the walk).** We call $G$ the group generated by $\iota_1$ and $\iota_2$ and we call $G_t$ the specialization of this group for a fixed value of $0 < t < 1$.

In the unweighted case, the algebraic nature of the generating series depends on whether $\sigma$ has finite or infinite order. More precisely, $G$ is finite if and only if the generating function is holonomic, i.e. satisfies a nontrivial linear differential equation with coefficients in $\mathbb{C}(x, y, t)$ in each of its three variables. On the other hand, when $G$ is infinite, $G_t$ can be either finite or infinite; see [15] for concrete examples. However, in that situation, the set of values of $t$ such that $G_t$ is finite is countable, see [8, Proposition 2.6].

**2.5. Uniformization of the curve.** In this section, we consider the uniformization problem in the genus one context, that has been solved in [14] for the case $t = 1$, and [11] for the case $0 < t < 1$. Let us consider a nondegenerate model of walk and assume that for all $t \in (0, 1)$, $\bar{E}_t$ is an elliptic curve. By Proposition 2.5, this corresponds to the situation where the step set is not included in any half-plane whose boundary passes through $(0, 0)$. By [11, Proposition 2.1], the elliptic curve $\bar{E}_t$ is biholomorphic to $\mathbb{C}/(\omega_1(t)\mathbb{Z} + \omega_2(t)\mathbb{Z})$ for some lattice $\omega_1(t)\mathbb{Z} + \omega_2(t)\mathbb{Z}$ of $\mathbb{C}$ via some $(\omega_1(t)\mathbb{Z} + \omega_2(t)\mathbb{Z})$-periodic holomorphic map $\Lambda$

$$ \Lambda: \mathbb{C} \to \bar{E}_t \\ \Lambda(\omega) := (x(\omega; t), y(\omega; t)), $$

where $x, y$ are rational functions of $\wp$ and its derivative $\partial_\omega \wp$, and $\wp$ is the Weierstrass function associated with the lattice $\omega_1(t)\mathbb{Z} + \omega_2(t)\mathbb{Z}$:

$$ \wp(\omega; t) = \frac{1}{\omega^2} + \sum_{(\ell_1, \ell_2) \in \mathbb{Z}^2 \setminus \{(0,0)\}} \left( \frac{1}{(\omega + \ell_1 \omega_1(t) + \ell_2 \omega_2(t))^2} - \frac{1}{(\ell_1 \omega_1(t) + \ell_2 \omega_2(t))^2} \right). $$
Then, the field of meromorphic functions on $\bar{E}_t$ is isomorphic to the field of meromorphic functions on $\mathbb{C}/(\omega_1(t)\mathbb{Z} + \omega_2(t)\mathbb{Z})$, that is itself isomorphic to the field of meromorphic functions on $\mathbb{C}$ that are $(\omega_1(t), \omega_2(t))$-periodic. For $t \in (0, 1)$ fixed, the latter field is equal to $\mathbb{C}(\wp, \partial_\omega \wp)$; see [23, Chapter 9, Theorem 1.8].

The maps $\iota_1, \iota_2$, and $\sigma$ may be analytically lifted to the $\omega$-plane $\mathbb{C}$ via the map $\Lambda^{-1}$. We denote these lifts by $\tilde{\iota}_1, \tilde{\iota}_2$, and $\tilde{\sigma}$ respectively. So we have the commutative diagrams

<figure><img src="image_115.png" /></figure>

For any $[x_0 : x_1]$ in $\mathbb{P}^1(\mathbb{C})$, we denote by $\Delta_1([x_0 : x_1]; t)$ the discriminant of the degree two homogeneous polynomial given by $y \mapsto \bar{K}(x_0, x_1, y; t)$. Let us write

$$ \Delta_1([x_0 : x_1]; t) = \sum_{i=0}^{4} \alpha_i(t) x_0^i x_1^{4-i}. $$

By [11, Theorem 1.11], the discriminant $\Delta_1([x_0 : x_1]; t)$ admits four distinct continuous real roots $a_1(t), \dots, a_4(t)$. They are numbered such that the cycle of $\mathbb{P}^1(\mathbb{R})$ starting from $-1$ to $\infty$ and from $-\infty$ to $-1$ crosses the $a_i$ in the order $a_1(t), \dots, a_4(t)$. Furthermore, $[1:0]$ is a root if and only if $\alpha_4(t) = 0$. In [11, Section 1.4], we see that $\alpha_4(t) = t^2(d_{1,0}^2 - 4d_{1,-1}d_{1,1})$. It follows that $[1:0]$ is a root of $\Delta_1([x_0 : x_1]; t)$ for one value of $t \in (0, 1)$, if and only if $[1:0]$ is a root of $\Delta_1([x_0 : x_1]; t)$ for all $t \in (0, 1)$.

Similarly, we denote by $b_1(t), \dots, b_4(t)$ the continuous real roots of the discriminant $x \mapsto \bar{K}(x, y_0, y_1; t)$, numbered in the same way, and we write $\Delta_2([y_0 : y_1]; t) = \sum_{i=0}^{4} \beta_i(t) y_0^i y_1^{4-i}$.

The following formulas have been proved

* in [14, Section 3.3] when $t = 1$,

* in [22] in the unweighted case,

* in [11, Proposition 2.1 and (2.16)], in the weighted case.

**Proposition 2.8** ([11], Proposition 2.1, Lemma 2.6, and (2.16)). For $i = 1, 2$, let us set $D_i(\star; t) := \Delta_i([\star: 1]; t)$. An explicit expression of the periods is given by the elliptic integrals

$$ \omega_1(t) = \mathbf{i} \int_{a_3(t)}^{a_4(t)} \frac{dx}{\sqrt{|D_1(x; t)|}} \in \mathbf{i}\mathbb{R}_{>0} \quad \text{and} \quad \omega_2(t) = \int_{a_4(t)}^{a_1(t)} \frac{dx}{\sqrt{D_1(x; t)}} \in \mathbb{R}_{>0}. $$

An explicit expression of the holomorphic map $\Lambda(\omega; t) = (x(\omega; t), y(\omega; t))$ is given by

* If $a_4(t) \neq [1:0]$, then $x(\omega; t) = [a_4(t) + \frac{D'_1(a_4(t);t)}{\wp(\omega;t)-\frac{1}{6}D''_1(a_4(t);t)} : 1]$;

* If $a_4(t) = [1:0]$, then $x(\omega; t) = [\wp(\omega; t) - \alpha_2(t)/3 : \alpha_3(t)]$;

* If $b_4(t) \neq [1:0]$, then $y(\omega; t) = [b_4(t) + \frac{D'_2(b_4(t);t)}{\wp(\omega-\omega_3(t)/2;t)-\frac{1}{6}D''_2(b_4(t);t)} : 1]$;

* If $b_4(t) = [1:0]$, then $y(\omega; t) = [\wp(\omega - \omega_3(t)/2; t) - \beta_2(t)/3 : \beta_3(t)]$.
An explicit expression of the involutions is given by

$$ \tilde{\iota}_1(\omega) = -\omega, \quad \tilde{\iota}_2(\omega) = -\omega + \omega_3 \quad \text{and} \quad \tilde{\sigma}(\omega) = \omega + \omega_3, $$

where

$$ \omega_3(t) = \int_{a_4(t)}^{X_{\pm}(b_4(t);t)} \frac{dx}{\sqrt{D_1(x;t)}} \in (0, \omega_2(t)), \quad (2.2) $$

and $X_{\pm}(y;t)$ are the two roots of $\bar{K}(X_{\pm}(y;t), y; t) = 0$.

**2.6. Meromorphic continuation of the generating function.** Let us summarize here the results of [11, Section 2.3]. Let us fix $t \in (0, 1)$. The generating function $Q(x, y; t)$ converges for $|x|, |y| < 1$. The projection of this set inside $\mathbb{P}^1(\mathbb{C}) \times \mathbb{P}^1(\mathbb{C})$ has a nonempty intersection with the kernel curve $\bar{E}_t$. In virtue of Lemma 2.1, we then find for $|x|, |y| < 1$ and $(x, y) \in \bar{E}_t$,

$$ 0 = K(x, 0; t)Q(x, 0; t) + K(0, y; t)Q(0, y; t) - K(0, 0; t)Q(0, 0; t) + xy. $$

To shorten several expressions hereafter, it is convenient to rewrite this equation introducing new auxiliary series $F_1$ and $F_2$:

$$ 0 = F_1(x; t) + F_2(y; t) - K(0, 0; t)Q(0, 0; t) + xy. \quad (2.3) $$

Since the series $F_1(x; t)$ and $F_2(y; t)$ converge for $|x|$ and $|y| < 1$ respectively, we then continue $F_1(x; t)$ for $(x, y) \in \bar{E}_t$ and $|y| < 1$ with the formula:

$$ F_1(x; t) = -F_2(y; t) + K(0, 0; t)Q(0, 0; t) - xy. $$

We continue $F_2(y; t)$ for $(x, y) \in \bar{E}_t$ and $|x| < 1$ similarly. There exists a connected set $O \subset \mathbb{C}$ such that

* $\Lambda(O) = \{(x, y) \in \bar{E}_t \text{ such that } |x| < 1 \text{ or } |y| < 1\}$;

* $\tilde{\sigma}^{-1}(O) \cap O \neq \emptyset$;

* $\bigcup_{\ell \in \mathbb{Z}} \tilde{\sigma}^{\ell}(O) = \mathbb{C}$.

There also exist meromorphic functions on $O$, $r_x(\omega; t)$ and $r_y(\omega; t)$, such that $r_x(\omega; t) = F_1(x(\omega; t); t)$ and $r_y(\omega; t) = F_2(y(\omega; t); t)$.

**Lemma 2.9 (Inclusion of poles).** The set of poles of $r_x(\omega; t)$ inside $O$ are contained in the set of poles of $x(\omega; t)$ with $|y(\omega; t)| < 1$. The set of poles of $r_y(\omega; t)$ inside $O$ are contained in the set of poles of $y(\omega; t)$ with $|x(\omega; t)| < 1$.

*Proof.* Let us use (2.3). On $O$, we have

$$ 0 = r_x(\omega; t) + r_x(\omega; t) - K(0, 0; t)Q(0, 0; t) + x(\omega; t)y(\omega; t). $$

Let us focus on $r_x(\omega; t)$, the proof for $r_y(\omega; t)$ is similar. Recall that $F_1(x; t)$ has no poles for $|x| < 1$. Since $r_x(\omega; t) = F_1(x(\omega; t); t)$, we find that $r_x(\omega; t)$ has no poles when $|x(\omega; t)| < 1$. With $\Lambda(O) = \{(x, y) \in \bar{E}_t ||x| < 1 \text{ or } |y| < 1\}$, we deduce that a pole of $r_x(\omega; t)$ inside $O$ satisfies $|y(\omega; t)| < 1$. We use $r_y(\omega; t) = F_2(y(\omega; t); t)$, and the fact that $F_2(y; t)$ has no poles for $|y| < 1$ to deduce that $r_y(\omega; t)$ has no poles when $|y(\omega; t)| < 1$. Therefore, the poles of $r_x(\omega; t)$ inside $O$ corresponds to the poles of $x(\omega; t)y(\omega; t)$ with $|y(\omega; t)| < 1$. The result follows. $\square$
With $\bigcup_{\ell \in \mathbb{Z}} \tilde{\sigma}^{\ell}(\mathcal{O}) = \mathbb{C}$ and $\tilde{\sigma}^{-1}(\mathcal{O}) \cap \mathcal{O} \neq \emptyset$, we then extend $r_x(\omega; t)$ and $r_y(\omega; t)$ as meromorphic functions on $\mathbb{C}$ where they satisfy the functional equations

$$r_x(\omega + \omega_3(t); t) = r_x(\omega; t) + b_x(\omega; t), \qquad (2.4)$$

$$r_x(\omega + \omega_1(t); t) = r_x(\omega; t), \qquad (2.5)$$

$$r_y(\omega + \omega_3(t); t) = r_y(\omega; t) + b_y(\omega; t),$$

$$r_y(\omega + \omega_1(t); t) = r_y(\omega; t), \qquad (2.6)$$

where $b_x(\omega; t) = y(-\omega; t)(x(\omega; t) - x(\omega + \omega_3(t); t))$ and $b_y(\omega; t) = x(\omega; t)(y(\omega; t) - y(-\omega; t))$.

From the functional equations (2.5) and (2.6), the set of poles of $\omega \mapsto r_x(\omega; t)$ and $\omega \mapsto r_y(\omega; t)$ are $\omega_1(t)$ periodic. With the other functional equations and $\bigcup_{\ell \in \mathbb{Z}} \tilde{\sigma}^{\ell}(\mathcal{O}) = \mathbb{C}$, we may deduce the expression of a discrete set containing the poles of $r_x$ and $r_y$.

**Lemma 2.10.** Let $P_x$ be the poles of $r_x$ in $\mathcal{O}$ and $P_{b,x}$ be the poles of $b_x$ in $\mathbb{C}$. The set of poles of $\omega \mapsto r_x(\omega; t)$ is included in $(P_x + \omega_3(t)\mathbb{Z}) \cup (P_{b,x} + \omega_3(t)\mathbb{Z})$. A similar statement holds for $r_y(\omega; t)$.

### 3. PRELIMINARY RESULTS ON DIFFERENTIAL ALGEBRAICITY

In this section, we prove some results on differential algebraicity, and more specifically on $\partial_t$-algebraicity of the functions that appear in Section 2.

Let us begin by definitions. Let $f(x_1, \dots, x_n)$ be a multivalued Puiseux series. For $i = 1, \dots, n$, we say that $f$ is $\partial_{x_i}$-algebraic if and only if it satisfies a nontrivial algebraic differential equation in the variable $x_i$, with coefficients in $\mathbb{Q}$. We say that $f$ is differentially algebraic in all its variables (or differentially algebraic for short) if and only if for all $1 \le i \le n$, $f$ is $\partial_{x_i}$-algebraic.

The following remark, proved e.g. in [18, Proposition 8, page 101], will be used several times in the sequel.

**Remark 3.1.** Let $f_1, \dots, f_n$ be differentially algebraic functions meromorphic on a common domain. A function satisfies a nontrivial algebraic differential equation with coefficients in $\mathbb{C}(f_1, \dots, f_n)$ if and only if it satisfies a nontrivial algebraic differential equation with coefficients in $\mathbb{Q}$.

The following lemma shows that the set of differentially algebraic functions is stable under many operations.

**Lemma 3.2 (Closure properties).** The set of differentially algebraic functions meromorphic on a domain is a field stable under derivations. If $f$ and $g$ are differentially algebraic and $f \circ g$ is well-defined then $f \circ g$ is differentially algebraic as well. If $f$ is differentially algebraic and admits an inverse $f^{-1}$, then $f^{-1}$ is also differentially algebraic.

*Proof.* See [8, Lemma 6.4] for the inverse property in the univariate case. The proof extends straightforwardly to the multivariate case. The rest of the statements follows from [2, Corollary 6.4 and Proposition 6.5]. □
In what follows, we might also consider functions of $t$ defined only on some intervals of $(0, 1)$. Let $\mathcal{D}$ be the field of multivalued functions that admit an expansion as convergent Puiseux series for all $t \in (0, 1)$, and that are differentially algebraic. In the sequel, when we will say that a function of $t$ defined (a priori) only of some intervals of $(0, 1)$ is differentially algebraic, it will be implicit that it may be continued as an element of $\mathcal{D}$.

The goal of the following results is to prove that various functions that appear in the uniformization of the elliptic curve are $\partial_t$-algebraic.

**Lemma 3.3** ([2, Lemma 6.10]). *The functions $\omega_1(t), \omega_2(t), \omega_3(t)$ belong to $\mathcal{D}$.<fnref n="1" /> Moreover, they are analytic on a complex neighborhood of $(0, 1)$.*

**Proposition 3.4.** *Functions of $\mathcal{D}(\wp(\omega; t), \partial_\omega \wp(\omega; t))$ are differentially algebraic in $t$ and $\omega$.*

*Proof.* Since the differentially algebraic functions form a field stable under the derivations (see Lemma 3.2), it suffices to show that $\wp(\omega; t)$ is differentially algebraic. It is well known that for $t \in (0, 1)$ fixed, $\wp(\omega; t)$ is $\partial_\omega$-algebraic. More precisely, it satisfies an equation of the form $(\partial_\omega \wp)^2 = 4\wp^3 - g_2(t)\wp - g_3(t)$, where $g_2(t), g_3(t)$ are the invariants of the elliptic curve. Differentiating with respect of $\omega$ allows us to eliminate the invariants, and obtain $\partial_\omega^3 \wp(\omega; t) = 12\wp(\omega; t)\partial_\omega \wp(\omega; t)$; see [1, (18.6.5)]. Hence $\wp(\omega; t)$ is $\partial_\omega$-algebraic.

Let us prove the $\partial_t$-algebraicity. In virtue of [2, Proposition 6.7], $\wp$ satisfies a nontrivial $\partial_t$-algebraic equation with coefficients in $\mathbb{C}(\omega_1(t), \omega_2(t))$. By Lemma 3.3, the periods $\omega_1(t)$ and $\omega_2(t)$ of $\wp$ are differentially algebraic, so in virtue of Remark 3.1, $\wp(\omega; t)$ is $\partial_t$-algebraic. $\square$

*Remark 3.5.* The same result holds with $\wp$ replaced by the Weierstrass function associated to the lattice $\omega_1(t)\mathbb{Z} + k\omega_2\mathbb{Z}$, or the lattice $\omega_1(t)\mathbb{Z} + \omega_3(t)\mathbb{Z}$.

**Definition 3.6** (Principal part). Let $f(\omega; t)$ be a meromorphic function at $\omega = a(t) \in \mathcal{D}$, with Laurent series $f(\omega; t) = \sum_{\ell=0}^{\infty} a_\ell(t)(\omega - a(t))^\ell$. The principal part of $f$ at $\omega = a(t)$ is the sum $\sum_{\ell=0}^{-1} a_\ell(t)(\omega - a(t))^\ell$ with the convention that it is 0 when $\nu \ge 0$. The coefficients of this principal part are $a_\nu(t), \dots, a_{-1}(t)$.

The following lemma will be used several times in the sequel.

**Lemma 3.7.** *The following statements hold:*

* Let $d(t) \in \mathcal{D}$ be an arbitrary function. We have $\wp(\omega; t) \in \mathcal{D}((\omega + d(t)));$

* $\wp(\omega; t) \in \omega^{-2}\mathcal{D}[[\omega]];$

* The coefficients of the principal parts of $\omega \mapsto \wp(\omega; t)$ belong to $\mathcal{D}$.

*Proof.* The last two assertions are straightforward consequences of the first one. Let us prove the first point. The function $d(t)$ and the poles of $\omega \mapsto \wp(\omega; t)$ are analytic on a convenient domain. So either $-d(t)$ is a pole of $\omega \mapsto \wp(\omega; t)$ with constant order with respect to $t$, or the set of $t$ such that $-d(t)$ is a pole of $\omega \mapsto \wp(\omega; t)$ is discrete. It follows that the order of the pole of $\omega \mapsto \wp(\omega; t)$ at $-d(t)$ is constant except on a discrete set. Since for $t$ fixed, $\omega \mapsto \wp(\omega; t)$ has pole of order at most two, we may write $\wp(\omega; t) = \sum_{\ell=k}^{\infty} c_\ell(t)(\omega + d(t))^\ell$.

<footnote n="1">¹They even are solutions of linear differential equations.</footnote>
Note that the coefficients $c_\ell(t)$ may have a pole when the order of the pole of $\omega \mapsto \wp(\omega; t)$ at $d(t)$ increases. In virtue of the field property of Lemma 3.2, combined with Proposition 3.4, we find that $(\omega + d(t))^{-k}\wp(\omega; t)$ is differentially algebraic. Note that $c_k(t)$ is the value at $-d(t)$ of the $\partial_t$-algebraic function $(\omega + d(t))^{-k}\wp(\omega; t)$. By the field property of Lemma 3.2 and Proposition 3.4, $(\omega + d(t))^{-k}\wp(\omega; t)$ is differentially algebraic in its two variables. By the composition property of Lemma 3.2 it follows that $c_k(t) \in \mathfrak{D}$. Let us fix $k \le n$ and assume that for $\ell = k, \dots, n$, $c_\ell(t) \in \mathfrak{D}$. Let us show that $c_{n+1}(t) \in \mathfrak{D}$. This will prove the result by induction. Let us define $h_n(\omega; t) = \wp(\omega, t) - \sum_{\ell=k}^n c_\ell(t)(\omega + d(t))^\ell$. By Proposition 3.4, the field property of Lemma 3.2, and the induction hypothesis, the function $t \mapsto h_n(\omega; t)$ is differentially algebraic in its two variables. Note that $c_{n+1}(t)$ is the value at $-d(t)$ of $(\omega + d(t))^{-(n+1)}h_n(\omega; t)$. By the composition property of Lemma 3.2 it follows that $c_k(t) \in \mathfrak{D}$. $\square$

As a consequence of what precedes, we deduce:

**Corollary 3.8.** The following holds:

* The functions $x(\omega; t)$ and $y(\omega; t)$ are differentially algebraic in their two variables;

* For $d(t) \in \mathfrak{D}$, we have $x(\omega; t), y(\omega; t) \in \mathfrak{D}((\omega + d(t))$;

* The poles and the coefficients of the principal parts of $\omega \mapsto x(\omega; t)$ and $\omega \mapsto y(\omega; t)$ belong to $\mathfrak{D}$.

*Proof.* We use the expressions of $x(\omega; t)$ and $y(\omega; t)$ given in Proposition 2.8. The elements involved in the expression are meromorphic on some complex neighborhood of $(0, 1)$ in the $t$-plane and are differentially algebraic by Proposition 3.4. Since the differentially algebraic elements form a field, see Lemma 3.2, the first point follows. Using Lemma 3.7, we deduce that $x(\omega; t), y(\omega; t) \in \mathfrak{D}((\omega + d(t)))$ for all $d(t) \in \mathfrak{D}$. Then the coefficients of the principal parts of $\omega \mapsto x(\omega; t)$ and $\omega \mapsto y(\omega; t)$ belong to $\mathfrak{D}$.

It remains to prove the differential algebraicity of the poles. Let $a(t)$ be a pole of $\omega \mapsto x(\omega; t)$ or $\omega \mapsto y(\omega; t)$. Then $a(t)$ is a continuous function solution of $\wp(a(t); t) = b(t)$, where $b(t)$ is $\partial_t$-algebraic.

Assume first that $\partial_\omega \wp(a(t); t)$ is identically zero or $a(t)$ is a pole of $\wp(\omega; t)$. By [23, page 270], this corresponds to the case where $a(t) \in \omega_1(t)\mathbb{Z}/2 + \omega_2(t)\mathbb{Z}/2$. By Lemma 3.3, $a(t)$ is meromorphic on a complex neighborhood of $(0, 1)$ and is $\partial_t$-algebraic. Then, it belongs to $\mathfrak{D}$.

Assume now that $\partial_\omega \wp(a(t); t)$ is not identically zero and $\wp(a(t); t) = b(t)$. Then, by the implicit function theorem, $a(t)$ admits an expansion as a meromorphic function on a complex neighborhood of any $t \in (0, 1)$ with $\partial_\omega \wp(a(t); t) \ne 0$. On that domain $\wp$ is locally invertible and its inverse is differentially algebraic in its two variables by Lemma 3.2. So we may write $\wp^{-1}(b(t); t) = a(t)$, where $\wp^{-1}$ is the local inverse of $\wp$. With the composition and inverse properties of Lemma 3.2, we deduce that $a(t)$ is $\partial_t$-algebraic. Furthermore, by the implicit function theorem, it admits an expansion as a convergent series on a complex neighborhood of any $t \in (0, 1)$. The set of $t$ such that $\partial_\omega \wp(a(t); t) \ne 0$ being dense, we find that the differential equation holds everywhere. This concludes the proof. $\square$
Recall, see Section 2.6, that

$$b_x(\omega; t) = y(-\omega; t)(x(\omega; t) - x(\omega + \omega_3(t); t)) \text{ and } b_y(\omega; t) = x(\omega; t)(y(\omega; t) - y(-\omega; t)).$$

**Corollary 3.9.** The following holds:

* The functions $b_x(\omega; t)$ and $b_y(\omega; t)$ are differentially algebraic in their two variables;

* For $d(t) \in \mathfrak{D}$, we have $b_x(\omega; t), b_y(\omega; t) \in \mathfrak{D}((\omega + d(t))$;

* The poles and the coefficients of the principal parts of $\omega \mapsto b_x(\omega; t)$ and $\omega \mapsto b_y(\omega; t)$ belong to $\mathfrak{D}$.

*Proof.* By Lemma 3.3, $\omega_3(t)$ belongs to $\mathfrak{D}$. This is now a straightforward application of Corollary 3.8, combined with the field property of Lemma 3.2. $\square$

Toward the proof of Theorem 1.1, we are going to face to many situations where the series is known to be $\partial_x$-algebraic (or $\partial_y$-algebraic) for all fixed values $t$. More precisely the differential algebraicity of the series will be proved to be equivalent to the existence of functions that are for all $t$ fixed, elliptic functions. Unfortunately, few things are known about the $t$-dependence of the coefficients. The following result will be the main ingredient in the proof of Theorem 1.1 since it gives a framework where we can state that the elliptic functions are differentially algebraic in all their variables.

**Theorem 3.10.** Let $\omega \mapsto f(\omega; t)$ be a function such that:

* For all $t \in (0, 1)$, $\omega \mapsto f(\omega; t) \in \mathbb{C}(\wp(\omega; t), \partial_\omega \wp(\omega; t))$.

* There are countably many elements of $\mathfrak{D}$, whose union forms the set of poles of $\omega \mapsto f(\omega; t)$.

* The coefficients of the principal parts of $\omega \mapsto f(\omega; t)$ are in $\mathfrak{D}$.

* There exists $a(t) \in \mathfrak{D}$ such that $f(a(t); t) \in \mathfrak{D}$.

Then, $f(\omega; t)$ is differentially algebraic in its two variables.

*Remark 3.11.* At first sight, nothing is explicitly assumed on the $t$-dependence of $t \mapsto f(\omega; t)$. However, the assumptions on the poles, on the principal parts, and on the special value $f(a(t); t)$, will imply that $t \mapsto f(\omega; t)$ is analytic on a convenient domain.

*Proof.* If $f$ is constant in the $\omega$ variable, then the result is clear. Assume that $f(\omega; t)$ is not constant. Let $a \in \mathbb{C}$. By the field property in Lemma 3.2, $f(\omega + a; t)$ satisfies the assumptions of Theorem 3.10. By the composition property, $f(\omega; t)$ is differentially algebraic in its two variables if and only if $f(\omega + a; t)$ is differentially algebraic in its two variables. Then, without loss of generality, we may reduce to the case where for any pole $b(t)$ of $\omega \mapsto f(\omega; t)$, $\partial_\omega \wp(b(t); t)$ is not identically zero. We may also assume that $a(t)$ is not a pole of $\omega \mapsto \wp(\omega; t)$.

Let us begin with the case where $\omega \mapsto f(\omega; t)$ is an even function. As we can see in the proof of [23, Lemma 1.9], we may write

$$f(\omega; t) = c(t) \prod_{i=1}^{\kappa_z} f_i(\omega; t) \prod_{j=1}^{\kappa_p} g_j(\omega; t),$$

where

* $c(t)$ is a function that does not depend upon $\omega$;

* $f_i(\omega; t)$ are of the form $\wp(\omega; t) - \wp(a(t); t)$, where $a(t)$ are zeros of $\omega \mapsto f(\omega; t)$;

* $g_j(\omega; t)$ are of the form $(\wp(\omega; t) - \wp(b(t); t))^{-1}$, where $b(t)$ are poles of $\omega \mapsto f(\omega; t)$.
Then, a partial fraction decomposition yields a sum of the form

$$f(\omega; t) = \tilde{c}(t) + \sum_{i=1}^{n_{\infty}} a_{i,\infty}(t) \wp(\omega; t)^i + \sum_j \sum_{i=1}^{n_j} \frac{a_{i,j}(t)}{(\wp(\omega; t) - \wp(b_j(t); t))^i} \quad (3.1)$$

By assumption, the $b_j(t)$ are differentially algebraic. Recall, see Lemma 3.7, that for all $j$, we have $\wp(\omega; t) \in \mathfrak{D}((\omega + b_j(t)))$ (resp. $\wp(\omega; t) \in \omega^{-2}\mathfrak{D}[[\omega]])$. Then, for every $i, j$,

$$\frac{a_{i,j}(t)}{(\wp(\omega; t) - \wp(b_j(t); t))^i} = \frac{a_{i,j}(t)}{(\partial_{\omega} \wp(b_j(t); t)(\omega - b_j(t)))^i} + O((\omega - b_j(t))^{-i+1}).$$

By the composition property of Lemma 3.2 and Proposition 3.4, for all $k, \ell$ the function $(\partial_{\omega}^k \wp(b_j(t); t))^{\ell}$ is differentially algebraic. Let us write the Taylor expansion of the function

$$f(\omega; t) = \sum_{i=-n_j}^{\infty} \tilde{a}_i(t) (\omega - b_j(t))^i.$$

Then, for $i < 0$, one has

$$\tilde{a}_i(t) = \frac{a_{i,j}(t)}{\partial_{\omega} \wp(b_j(t); t)^i} + f_{i,j}, \text{ where } f_{i,j} \in \mathfrak{D}(a_{i+1,j}(t), \dots, a_{n_j,j}(t)).$$

Since the coefficients of the principal part at $b_j(t)$ are differentially algebraic we have $\tilde{a}_i(t) \in \mathfrak{D}$. By Lemma 3.2, $\mathfrak{D}$ is a field, and we find by a decreasing induction that for all $1 \le i \le n_j$, $a_{i,j}(t) \in \mathfrak{D}$. Similarly, for all $i$, we have

$$a_{i,\infty}(t) \wp(\omega; t)^i = \omega^{-2i} a_{i,\infty}(t) + O(\omega^{-2i+1}).$$

Then the coefficient of the term in $\omega^{-2i}$ with $i > 0$ is of the form $a_{i,\infty}(t) + f_i$, where $f_i \in \mathfrak{D}(a_{i+1,\infty}(t), \dots, a_{n_{\infty},\infty}(t))$. Since the coefficients of the principal part at 0 are differentially algebraic, we find $a_{i,\infty}(t) + f_i \in \mathfrak{D}$. By Lemma 3.2, $\mathfrak{D}$ is a field, and we find by a decreasing induction that for all $1 \le i \le n_{\infty}$, $a_{i,\infty}(t) \in \mathfrak{D}$. Recall that by assumption, $f(a(t); t)$ is $\partial_t$-algebraic. By Lemma 3.2 and Proposition 3.4, we find

$$\tilde{d}(t) := \sum_{i=1}^{n_{\infty}} a_{i,\infty}(t) \wp(a(t); t)^i + \sum_j \sum_{i=1}^{n_j} \frac{a_{i,j}(t)}{(\wp(a(t); t) - \wp(b_j(t); t))^i} \in \mathfrak{D}.$$

By the subtraction property of Lemma 3.2 we deduce that $\tilde{c}(t) = f(a(t); t) - \tilde{d}(t)$ is $\partial_t$-algebraic. In virtue of Lemma 3.2 and Proposition 3.4, every term in the right-hand side of (3.1) is differentially algebraic. With the field property of Lemma 3.2, this concludes the proof in the even case.

Assume that $\omega \mapsto f(\omega; t)$ is odd. The function $\partial_{\omega} \wp(\omega; t)^{-1} f(\omega; t)$ is even, and $\omega_1(t)\mathbb{Z} + \omega_2(t)\mathbb{Z}$, the poles of $\partial_{\omega} \wp(\omega; t)$, are $\partial_t$-algebraic; see Lemma 3.3. Then, we may apply the even case to deduce that $f(\omega; t)$ is of the form

$$\partial_{\omega} \wp(\omega; t) \tilde{c}(t) + \sum_{i=1}^{n_{\infty}} a_{i,\infty}(t) \partial_{\omega} \wp(\omega; t) \wp(\omega; t)^i + \sum_j \sum_{i=1}^{n_j} \frac{a_{i,j}(t) \partial_{\omega} \wp(\omega; t)}{(\wp(\omega; t) - \wp(b_j(t); t))^i}.$$
Setting $a_{0,\infty}(t) := \tilde{c}(t)$, we may rewrite the latter expression as

$$ \sum_{i=0}^{n_{\infty}} a_{i,\infty}(t) \partial_{\omega} \wp(\omega; t) \wp(\omega; t)^i + \sum_{j} \sum_{i=1}^{n_j} \frac{a_{i,j}(t) \partial_{\omega} \wp(\omega; t)}{(\wp(\omega; t) - \wp(b_j(t); t))^i} $$

By Proposition 3.4, for all $j$, we have $\partial_{\omega}\wp(\omega; t) \in \mathfrak{D}((\omega - b_j(t)))$ (resp. we have $\partial_{\omega}\wp(\omega; t) \in \mathfrak{D}((\omega)))$. The same reasoning as in the even case shows that for all $1 \le i \le n_j$, the functions $a_{i,j}(t)$ are differentially algebraic. Similarly, for all $0 \le i \le n_{\infty}$, the functions $a_{i,\infty}(t)$ are differentially algebraic. By Proposition 3.4 and Lemma 3.2, we find that $f(\omega; t)$ is differentially algebraic. This completes the proof in the odd case.

Let us consider the general case. Note that by Proposition 3.4, $\wp(\omega; t) - \wp(a(t); t)$ is differentially algebraic. So for all $n$, Lemma 3.2 ensures that $f(\omega; t)$ is differentially algebraic if and only if $(\wp(\omega; t) - \wp(a(t); t))^n f(\omega; t)$ is differentially algebraic. So without loss of generality, we may reduce to the case where $f(\pm a(t); t) = 0$. We write $f(\omega; t) = f_+(\omega; t) + f_-(\omega; t)$, where

$$ f_+(\omega; t) := \frac{f(\omega; t) + f(-\omega; t)}{2}, $$

$$ f_-(\omega; t) := \frac{f(\omega; t) - f(-\omega; t)}{2}. $$

The poles of $\omega \mapsto f_{\pm}(\omega; t)$ are poles of $f$ or opposite of the latter. By Lemma 3.2, they are $\partial_t$-algebraic and the coefficients of the principal parts are in $\mathfrak{D}$. Since $f(\pm a(t); t) = 0$ we find $f_{\pm}(a(t); t) = 0$. In particular it is differentially algebraic. From the even and the odd cases, $f_{\pm}(\omega; t)$ are differentially algebraic in their two variables. Since the sum of two differentially algebraic functions is differentially algebraic, see Lemma 3.2, we deduce that $f(\omega; t) = f_+(\omega; t) + f_-(\omega; t)$ is differentially algebraic. $\square$

Remark 3.12.

* As in Remark 3.5, we may consider $\tilde{\wp}(\omega; t)$, the Weierstrass functions associated to the lattice $\omega_1(t)\mathbb{Z} + \omega_3(t)\mathbb{Z}$, or the lattice $\omega_1(t)\mathbb{Z} + k\omega_2(t)\mathbb{Z}$, with $k \in \mathbb{N}^*$. Then, the proof of Theorem 3.10 can be straightforwardly adapted to this new lattice. We then deduce the following. If $\omega \mapsto f(\omega; t)$ is a function such that:

(1) For all $t \in (0, 1)$, $\omega \mapsto f(\omega; t) \in \mathbb{C}(\tilde{\wp}(\omega; t), \partial_{\omega}\tilde{\wp}(\omega; t))$.

(2) There are countably many elements of $\mathfrak{D}$, whose union forms the set of poles of $\omega \mapsto f(\omega; t)$.

(3) The coefficients of the principal parts of $\omega \mapsto f(\omega; t)$ are in $\mathfrak{D}$.

(4) There exists $a(t) \in \mathfrak{D}$ such that $f(a(t); t) \in \mathfrak{D}$.

Then, $f(\omega; t)$ is differentially algebraic in its two variables.

* Let us now just assume that $\omega \mapsto f(\omega; t)$ satisfies the above first three points and let $a(t) \in \mathfrak{D}$ that is not a pole. Then, $f(\omega; t) - f(a(t); t)$ satisfies the four points and is therefore differentially algebraic. By construction, the function $f(\omega; t) - f(a(t); t)$ has the same principal parts as $f(\omega; t)$.

Although $r_x$ and $r_y$ are not elliptic functions, we will see in the next section that it is sufficient to control the behavior of their poles and coefficients in order to apply Theorem 3.10.
**Lemma 3.13.** The following holds:

(A1) The poles and coefficients of the principal parts of $\omega \mapsto r_x(\omega; t)$ belong to $\mathfrak{D}$.

(A2) There exists $a(t) \in \mathfrak{D}$ such that $r_x(a(t); t) \in \mathfrak{D}$.

Similar statements hold for $r_y$.

*Proof.* Let us prove the result for $r_x$, the reasoning for $r_y$ is similar. We refer to Section 2.6 for the notations used in this proof.

Recall that the series $Q(x, y; t)$ converges for $|x|, |y|, |t| < 1$. Let us consider $t$ in $(0, 1)$. Take $\omega \in \mathcal{O}$ (note that $\mathcal{O}$ depends continuously on $t$), for each of the domains $|x(\omega; t)| < 1$ and $|y(\omega; t)| < 1$, one has the following equality of functions:

$$F_1(x(\omega; t); t) = r_x(\omega; t) \quad \text{and} \quad F_2(y(\omega; t); t) = r_y(\omega; t),$$

with no poles on these domains. Via the equality $0 = r_x(\omega; t)+r_y(\omega; t)-K(0, 0; t)Q(0, 0; t)+x(\omega; t)y(\omega; t)$, and Lemma 2.9 on the inclusion of poles, we deduce that the poles inside $\mathcal{O}$ of $\omega \mapsto r_x(\omega; t)$ are the poles inside $\mathcal{O}$ of $\omega \mapsto x(\omega; t)y(\omega; t)$ with $|y(\omega; t)| < 1$. What is more, on that domain, $\omega \mapsto x(\omega; t)y(\omega; t)$ and $\omega \mapsto r_x(\omega; t)$ have the same principal parts. By Corollary 3.8, the poles of $\omega \mapsto r_x(\omega; t)$ inside $\mathcal{O}$ are differentially algebraic. Furthermore, the corresponding principal parts have differentially algebraic coefficients.

Recall, see (2.4), that $r_x(\omega + \omega_3(t); t) = r_x(\omega; t) + b_x(\omega; t)$. By Corollary 3.9, the poles and the coefficients of the principal parts of $\omega \mapsto b_x(\omega; t)$ are differentially algebraic. By Lemma 3.3, $\omega_3(t)$ is differentially algebraic. Recall that $\bigcup_{\ell \in \mathbb{Z}} \tilde{\sigma}^\ell(\mathcal{O}) = \mathbb{C}$. With (2.4) and what precedes, we get assertion (A1).

It remains to prove assertion (A2). To lighten the notations we omit the dependence in $t$ in what follows. Let us write $K(x, y; t) = \tilde{B}_0(y)+x\tilde{B}_1(y)+x^2\tilde{B}_2(y)$. Let $\omega_0(t) \in \mathcal{O}$ such that

$$y(\omega_0) = 0 \quad \text{and} \quad x(\omega_0) = \frac{-\tilde{B}_1(y(\omega_0)) + \sqrt{\tilde{B}_1(y(\omega_0))^2 - 4\tilde{B}_0(y(\omega_0))\tilde{B}_2(y(\omega_0))}}{2\tilde{B}_2(y(\omega_0))}.$$

The $y$-valuation of $\tilde{B}_2(y)$ being at most two, we consider the following subcases.

* If it is 0 or 1, the valuation of the algebraic function $y \times \frac{-\tilde{B}_1(y)+\sqrt{\tilde{B}_1(y)^2-4\tilde{B}_0(y)\tilde{B}_2(y)}}{\tilde{B}_2(y)}$ is nonnegative and then $\omega_0$ is not a pole of $x(\omega; t)y(\omega; t)$.

* If it is 2, then $4\tilde{B}_0(y)\tilde{B}_2(y)$ converges to 0 when $y$ goes to 0 and hence the same holds for $-\tilde{B}_1(y) + \sqrt{\tilde{B}_1(y)^2 - 4\tilde{B}_0(y)\tilde{B}_2(y)}$.

We further find that $y \times (-\tilde{B}_1(y) + \sqrt{\tilde{B}_1(y)^2 - 4\tilde{B}_0(y)\tilde{B}_2(y)}) \in O(y^2)$. In that case, we find that $\omega_0$ is not a pole of $x(\omega; t)y(\omega; t)$ either. With $K(0, 0; t)Q(0, 0; t) = F_2(y(\omega_0; t); t) = r_y(\omega_0; t)$, and $0 = r_x(\omega; t) + r_y(\omega; t) - K(0, 0; t)Q(0, 0; t) + x(\omega; t)y(\omega; t)$, we then find $0 = r_x(\omega_0; t) + x(\omega_0; t)y(\omega_0; t)$. It then suffices to show that $x(\omega_0; t)y(\omega_0; t)$ is differentially algebraic. With the expression of $y(\omega_0; t)$ in Proposition 2.8, we find that $\omega_0$ is solution of an equation of the form $\wp(\omega_0; t) = b(t)$ with $b(t) \in \mathfrak{D}$. With the same reasons as in the proof of Corollary 3.8, we find that $\omega_0$ is differentially algebraic, and $x(\omega_0; t), y(\omega_0; t) \in \mathfrak{D}$. Then $r_x(\omega_0(t); t) \in \mathfrak{D}$. This concludes the proof. $\square$