# A Fuss-type family of positive definite sequences

Wojciech Mlotkowski, Karol A. Penson

► To cite this version:

Wojciech Mlotkowski, Karol A. Penson. A Fuss-type family of positive definite sequences. Colloquium Mathematicum, 2018, 151 (2), pp.289 - 304. 10.4064/cm6894-2-2017. hal-01744071

HAL Id: hal-01744071

https://hal.sorbonne-universite.fr/hal-01744071

Submitted on 27 Mar 2018

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# A FUSS-TYPE FAMILY OF POSITIVE DEFINITE SEQUENCES

BY

WOJCIECH MŁOTKOWSKI (Wrocław) and KAROL A. PENSON (Paris)

**Abstract.** We study a two-parameter family $a_n(p, t)$ of deformations of the Fuss numbers. We show a sufficient condition for positive definiteness of $a_n(p, t)$ and prove that some of the corresponding probability measures are infinitely divisible with respect to the additive free convolution.

**1. Introduction.** The aim of the paper is to study a two-parameter family of sequences

$$ (1.1) \qquad a_n(p, t) := \binom{np}{n} \frac{n(2p - t - pt) + 2}{(np - n + 1)(np - n + 2)}, \quad p, t \in \mathbb{R} $$

which can be regarded as deformation of the Fuss numbers. Assuming that $p \ge 0$ we prove that the sequence $a_n(p, t)$ is positive definite if and only if $p \ge 1$ and $g(p) \le t \le 2p/(p+1)$, where $g(p)$ is defined by (3.9). We conjecture that the assumption $p \ge 0$ is redundant.

The case $t = 2p/(p+1)$ is particularly interesting due to connections with the work [6] of M. Bousquet-Mélou and G. Schaeffer. They introduced the notion of constellation as a tool for studying factorization problems in symmetric groups. For $p \ge 2$ a *p-constellation* is a 2-cell decomposition of the oriented sphere into vertices, edges and faces, with faces colored black and white in such a way that:

* all faces adjacent to a given white face are black and vice versa,

* the degree of any black face is $p$,

* the degree of any white face is a multiple of $p$.

A constellation is called *rooted* if one of the edges is distinguished.

The number of rooted *p*-constellations formed of *n* polygons, counted up to isomorphism, is given by

$$ (1.2) \qquad C_p(n) := \binom{np}{n} \frac{(p+1)p^{n-1}}{(np-n+1)(np-n+2)}, \quad p \ge 2, n \ge 1 $$

<footnote n="2010">2010 Mathematics Subject Classification: Primary 44A60; Secondary 33C20, 46L54.</footnote>

<footnote>Key words and phrases: Fuss numbers, Mellin convolution, Meijer G-function, free probability.</footnote>

<footnote>Received 11 February 2016; revised 9 February 2017.</footnote>

<footnote>Published online 12 January 2018.</footnote>
(see [6, Corollary 2.4]). Some of these sequences appear in the On-Line Encyclopedia of Integer Sequences (OEIS) [27], namely: $C_2 = A000257$, $C_3 = A069726$, $C_4 = A090374$.

We will prove that the probability distribution $\eta(p, t)$ corresponding to a positive definite sequence $a_n(p, t)$ is absolutely continuous, except for $\eta(1, 1) = \delta_1$, and the support of $\eta(p, t)$ is $[0, p^p(p-1)^{1-p}]$. The density function will be denoted $f_{p,t}(x)$. For $p=2$ and $p=3$ we compute the $R$-transform of $\eta(p, t)$. We prove that $\eta(2, t)$ (resp. $\eta(3, t)$) is infinitely divisible with respect to the additive free convolution if and only if $1 \le t \le 4/3$ (resp. $1/2 \le t \le 3/2$). The general problem of determining for which parameters $p, t$ the distribution $\eta(p, t)$ is infinitely divisible with respect to the additive free convolution remains open.

Finally, let us record some other sequences from OEIS which are related to this work: A005807: $2a_n(2, 1/2)$ (sums of adjacent Catalan numbers), A007226: $2a_n(3, 1/2)$ (studied in [15]), A007054: $3a_n(2, 4/3)$ (super ballot numbers), A038629: $3a_n(2, 2/3)$, A000139: $2a_n(3, 3/2)$, A197271: $5a_n(4, 8/5)$, A197272: $3a_n(5, 5/3)$. In Section 4 we also encounter sequences A022558 and A220910.

**2. Fuss numbers.** The Fuss-Catalan numbers $\binom{np+1}{n} \frac{1}{np+1}$ have several combinatorial applications [9, 7, 2, 25, 6, 24]. They count for example:

1. the number of ways of dividing a convex polygon with $n(p-1)+2$ vertices into $n$ disjoint $(p+1)$-gons by means of nonintersecting diagonals,

2. the number of sequences $(a_1, \dots, a_{np})$, where $a_i \in \{1, 1-p\}$, with all partial sums $a_1 + \dots + a_k$ nonnegative and with $a_1 + \dots + a_{np} = 0$,

3. the number of noncrossing partitions $\pi$ of $\{1, 2, \dots, n(p-1)\}$, such that $p-1$ divides the cardinality of every block of $\pi$,

4. the number of $p$-cacti formed of $n$ polygons [6].

The generating function

$$ (2.1) \qquad \mathcal{B}_p(z) := \sum_{n=0}^{\infty} \binom{np+1}{n} \frac{z^n}{np+1} $$

satisfies

$$ (2.2) \qquad \mathcal{B}_p(z) = 1 + z\mathcal{B}_p(z)^p. $$

Recall also Lambert's formula for the Taylor expansion of the powers of $\mathcal{B}_p(z)$:

$$ (2.3) \qquad \mathcal{B}_p(z)^r = \sum_{n=0}^{\infty} \binom{np+r}{n} \frac{rz^n}{np+r}. $$

These formulas remain true for $p, r \in \mathbb{R}$ and the coefficients $\binom{np+r}{n} \frac{r}{np+r}$
(understood to be 1 for $n = 0$ and $\frac{r}{n!} \prod_{i=1}^{n-1} (np + r - i)$ for $n \ge 1$) are called two-parameter Fuss numbers or Raney numbers [9, 13, 22, 12, 8].

In some cases the function $B_p$ can be written explicitly, for example

$$ 
\begin{aligned}
B_2(z) &= \frac{2}{1 + \sqrt{1 - 4z}} = \frac{1 - \sqrt{1 - 4z}}{2z}, \\
B_3(z) &= \frac{3}{3 - 4 \sin^2 \alpha}, \\
B_{3/2}(z) &= \frac{3}{(\sqrt{3} \cos \beta - \sin \beta)^2},
\end{aligned}
$$

where $\alpha = \frac{1}{3} \arcsin(\sqrt{27z/4})$, $\beta = \frac{1}{3} \arcsin(3z\sqrt{3}/2)$ [16].

Fuss numbers also have applications in free probability and in the theory of random matrices, as moments of the multiplicative free powers of the Marchenko-Pastur distribution [1, 3, 13, 14, 18]. This implies that for $p \ge 1$ the sequence $\binom{np+1}{n} \frac{1}{np+1}$ is positive definite. More generally, the sequence $\binom{np+r}{n} \frac{r}{np+r}$ is positive definite if and only if either $p \ge 0$, $0 \le r \le p$, or $p \le 0$, $p-1 \le r \le 0$ or $r = 0$ [11, 13, 16, 12, 8, 21]. The case $r = 0$ is trivial, as it gives the sequence $(1, 0, 0, 0, \dots)$, the moments of $\delta_0$. The distributions corresponding to the second case, $p \le 0$, $p-1 \le r \le 0$, are just reflections of those corresponding to $p \ge 0$, $0 \le r \le p$. This is a consequence of the identity

$$ (2.4) \qquad \binom{np+r}{n} \frac{r(-1)^n}{np+r} = \binom{n(1-p)-r}{n} \frac{-r}{n(1-p)-r}. $$

For $p > 1$ and $r > 0$ we have the integral representation

$$ \binom{np+r}{n} \frac{r}{np+r} = \int_{0}^{c(p)} x^n W_{p,r}(x) dx, $$

where $c(p) := p^p(p-1)^{1-p}$, and $W_{p,r}$ is given by

$$ (2.5) \qquad W_{p,r}(x) = \frac{(\sin(p-1)\phi)^{p-r-1} \sin\phi \sin r\phi}{\pi(\sin p\phi)^{p-r}}, $$

where

$$ (2.6) \qquad x = \rho(\phi) = \frac{(\sin p\phi)^p}{\sin\phi (\sin(p-1)\phi)^{p-1}}, \quad 0 < \phi < \pi/p. $$

This function is nonnegative if and only if $r \le p$ [10, 18, 8].

If $p = k/l$ is a rational number, $1 \le l < k$, then $W_{p,r}$ can be expressed in terms of the Meijer G-function [22, 17]

$$ (2.7) \qquad W_{p,r}(x) = \frac{rp^r}{x(p-1)^{r+1/2}\sqrt{2k\pi}} G_{k,k}^{k,0} \left( \frac{x^l}{c(p)^l} \middle| \begin{matrix} \alpha_1, \dots, \alpha_k \\ \beta_1, \dots, \beta_k \end{matrix} \right), $$
where $x \in (0, c(p))$ and the parameters $\alpha_j, \beta_j$ are given by

$$ (2.8) \qquad \alpha_j = \begin{cases} \frac{j}{l} & \text{if } 1 \le j \le l, \\ \frac{r+j-l}{k-l} & \text{if } l+1 \le j \le k, \end{cases} $$

$$ (2.9) \qquad \beta_j = \frac{r+j-1}{k}, \quad 1 \le j \le k. $$

EXAMPLES. Let us record formulas for the functions $W_{p,r}$ for $p = 2, 3, 3/2$ and $r = 1, 2$. In these cases $W_{p,r}$ can be expressed as an elementary function [21, 22, 17]:

$$ (2.10) \qquad W_{2,1}(x) = \frac{1}{2\pi} \sqrt{\frac{4-x}{x}}, $$

$$ (2.11) \qquad W_{2,2}(x) = \frac{1}{2\pi} \sqrt{x(4-x)}, $$

where $x \in (0, 4)$. $W_{2,1}$ is the density of the Marchenko-Pastur distribution and $W_{2,2}$ is Wigner's semicircle law translated by 2. Moreover,

$$ (2.12) \qquad W_{3,1}(x) = \frac{3\left(1 + \sqrt{1 - \frac{4x}{27}}\right)^{2/3} - (4x)^{1/3}}{3^{1/2}\pi(4x)^{2/3}\left(1 + \sqrt{1 - \frac{4x}{27}}\right)^{1/3}}, $$

$$ (2.13) \qquad W_{3,2}(x) = \frac{9\left(1 + \sqrt{1 - \frac{4x}{27}}\right)^{4/3} - (4x)^{2/3}}{2\pi 3^{3/2}(4x)^{1/3}\left(1 + \sqrt{1 - \frac{4x}{27}}\right)^{2/3}}, $$

where $x \in (0, 27/4)$, and

$$ (2.14) \qquad W_{3/2,1}(x) = 3^{1/2} \frac{\left(1 + \sqrt{1 - \frac{4x^2}{27}}\right)^{1/3} - \left(1 - \sqrt{1 - \frac{4x^2}{27}}\right)^{1/3}}{2(2x)^{1/3}\pi} + 3^{1/2} \frac{\left(1 + \sqrt{1 - \frac{4x^2}{27}}\right)^{2/3} - \left(1 - \sqrt{1 - \frac{4x^2}{27}}\right)^{2/3}}{4\pi}, $$

$$ (2.15) \qquad W_{3/2,2}(x) = \frac{3^{1/2}(2x)^{5/3}}{8\pi} \left( \left(1 + \sqrt{1 - \frac{4x^2}{27}}\right)^{1/3} - \left(1 - \sqrt{1 - \frac{4x^2}{27}}\right)^{1/3} \right) + \frac{3^{1/2}(2x)^{1/3}(x^2 - 1)}{4\pi} \left( \left(1 + \sqrt{1 - \frac{4x^2}{27}}\right)^{2/3} - \left(1 - \sqrt{1 - \frac{4x^2}{27}}\right)^{2/3} \right), $$

where $x \in (0, 3\sqrt{3}/2)$. The function $W_{3/2,2}(x)$ is not nonnegative on its domain.
**3. A family of sequences.** For $p, t \in \mathbb{R}$ define $a_n(p, t)$ as an affine combination of $inom{np+1}{n} \frac{1}{np+1}$ and $inom{np+2}{n} \frac{2}{np+2}$:

$$
(3.1) \qquad a_n(p,t) := \binom{np+1}{n} \frac{t}{np+1} + \binom{np+2}{n} \frac{2(1-t)}{np+2}
$$

$$
(3.2) \qquad = \binom{np}{n} \frac{n(2p - t - pt) + 2}{(np - n + 1)(np - n + 2)},
$$

in particular $a_0(p,t) = 1$.

The generating function is

$$
(3.3) \qquad t\mathcal{B}_p(z) + (1-t)\mathcal{B}_p(z)^2 = \sum_{n=0}^{\infty} a_n(p,t)z^n.
$$

For example,

$$
t\mathcal{B}_2(z) + (1-t)\mathcal{B}_2(z)^2 = \frac{1-t+3tz-2z-(1-t+tz)\sqrt{1-4z}}{2z^2},
$$

$$
t\mathcal{B}_3(z) + (1-t)\mathcal{B}_3(z)^2 = \frac{9 - 12t \sin^2 \alpha}{(3 - 4 \sin^2 \alpha)^2},
$$

$$
t\mathcal{B}_{3/2}(z) + (1-t)\mathcal{B}_{3/2}(z)^2 = \frac{9 - 6t \sin^2 \beta + 6t\sqrt{3} \sin \beta \cos \beta}{(\sqrt{3} \cos \beta - \sin \beta)^4}
$$

$$
\text{where } \alpha = \frac{1}{3} \arcsin(\sqrt{27z/4}), \beta = \frac{1}{3} \arcsin(3z\sqrt{3}/2).
$$

We are going to study positive definiteness of $a_n(p, t)$. First we observe

PROPOSITION 3.1. If the sequence $a_n(p, t)$ is positive definite then

$$
(3.4) \qquad 2p - pt - t^2 + 3t - 3 \ge 0.
$$

In particular $t \neq 2$ and either $p \leq -3$ or $p \geq 1$.

Proof. The left hand side is just $a_2(p,t) - a_1(p,t)^2$. ■

EXAMPLES. 1. For $p=1$ we have $a_n(1,t) = 1+n-nt$. Since

$$
a_2(1, t) - a_1(1, t)^2 = -(t-1)^2,
$$

the sequence $a_n(1, t)$ is positive definite if and only if $t = 1$. Note that $a_n(1, 1) = 1$ is the moment sequence of the one-point measure $\delta_1$.

2. For $t = 2/(p+1)$ we get

$$
a_n(p, 2/(p+1)) = \binom{np}{n} \frac{2}{np - n + 2}.
$$

If $p > 1$ then this is a product of two positive definite sequences: $\binom{np}{n}$ (see
$[16, 26]$) and $2/(np - n + 2)$.
3. Similarly, for $p > 1$ and $t = 2p/(p+1)$ the sequence

$$a_n(p, 2p/(p+1)) = \binom{np}{n} \frac{2}{(np-n+1)(np-n+2)}.$$

is positive definite. Note that from (1.2) we have

$$(3.5) \qquad C_p(n) = \frac{(p+1)p^n}{2p} a_n \left( p, \frac{2p}{p+1} \right),$$

so for $p \ge 1$ the sequence $C_p(n)$ is positive definite.

As already noted, the sequence $a_n(p, t)$ is an affine combination of $\binom{np+1}{n} \frac{1}{np+1}$ and $\binom{np+2}{n} \frac{2}{np+2}$. The former is positive definite for $p \ge 1$ and the latter for $p \ge 2$. This implies that $a_n(p, t)$ is positive definite for $p \ge 2$ and $0 \le t \le 1$. We are going to prove something stronger. Note that if $t_1 \le t_2 \le t_3$ and the sequences $a_n(p, t_1)$, $a_n(p, t_3)$ are positive definite then so is $a_n(p, t_2)$ as their convex combination.

If we assume that $p > 1$ then

$$a_n(p,t) = \int_0^{c(p)} x^n f_{p,t}(x) dx,$$

where

$$f_{p,t}(x) = tW_{p,1}(x) + (1-t)W_{p,2}(x).$$

Then the positive definiteness of $a_n(p, t)$ is equivalent to $f_{p,t}$ being nonnegative on $(0, c(p))$. For example, the function

$$(3.6) \qquad f_{2,t}(x) = \frac{t+x-tx}{2\pi} \sqrt{\frac{4-x}{x}}$$

is nonnegative on $(0, 4)$ if and only if $0 \le t \le 4/3$.

<figure><img src="image_2.png" /><figcaption>Fig. 1. The density function $f_{3/2,1/5}(x)$</figcaption></figure>
By (2.5) we can write

$$ (3.7) \quad f_{p,t}(x) = \frac{\sin^2 \phi (\sin (p-1)\phi)^{p-3} [t \sin (p-1)\phi + 2(1-t) \sin p\phi \cos \phi]}{\pi (\sin p\phi)^{p-1}} $$

for $x$ as in (2.6). Define

$$ \begin{align} (3.8) \quad \Psi_{p,t}(\phi) &= t \sin (1 - 1/p)\phi + 2(1 - t) \sin \phi \cos \phi/p \\ &= (2 - t) \sin \phi \cos \phi/p - t \cos \phi \sin \phi/p \\ &= (1 - t) \sin (1 + 1/p)\phi + \sin (1 - 1/p)\phi. \end{align} $$

Then the sequence $a_n(p, t)$ is positive definite if and only if $\Psi_{p,t}(\phi) \ge 0$ for $\phi \in [0, \pi]$. For $p \ge 1$ set

$$ (3.9) \quad g(p) := \min\{t \in \mathbb{R} : \Psi_{p,t}(\phi) \ge 0 \text{ for all } 0 < \phi < \pi\}. $$

Since $\Psi_{p,t}(\pi) = t \sin(\pi/p)$ and $\Psi_{p,1}(\phi) = \sin(1-1/p)\phi$, we have $0 \le g(p) \le 1$ for all $p \ge 1$.

**PROPOSITION 3.2.** The function $g$ is continuous on $[1, \infty)$, strictly decreasing on $[1, 2]$, $g(p) = 0$ for $p \ge 2$ and $g(1) = 1$, $g(3/2) = 1/5$.

*Proof.* For $p=1$ we have $\Psi_{1,t}(\phi) = (1-t)\sin 2\phi$, which implies $g(1)=1$. If $p \ge 2$ then $\Psi_{p,0}(\phi) = 2\sin\phi\cos\phi/p$ is nonnegative for $\phi \in [0, \pi]$, which yields $g(p) = 0$.

Now observe that for fixed $t, \phi$ with $0 \le t \le 1$, $0 < \phi \le \pi$, the function $p \mapsto \Psi_{p,t}(\phi)$ is strictly increasing on $[1, 2]$. Indeed, we can write

$$ \Psi_{p,t}(\phi) = 2(1-t) \sin \phi \cos \phi/p + t \sin (\phi - \phi/p), $$

and if $0 < \phi \le \pi$ then both the summands are increasing with $p \in [1, 2]$. This implies that $g(p)$ is strictly decreasing on $[1, 2]$.

To prove continuity of $g$ assume that $1 \le p_1 < p_2 \le 2$ and set $t_1 := g(p_1)$ and $t_2 := g(p_2)$. Then $t_1 > t_2$, $\Psi_{p_1,t_1}(\phi) \ge 0$ for all $\phi \in [0, \pi]$ and there is $\phi_1$ with $p_1\pi/(1+p_1) < \phi_1 < \pi$ such that $\Psi_{p_1,t_1}(\phi_1) = 0$. Then $\Psi_{p_2,t_1}(\phi) > 0$ for all $\phi \in (0, \pi]$. From the third expression in (3.8) we have

$$ -c_1 := \sin (1 + 1/p_1)\phi_1 < 0. $$

If we assume that $(p_2 - p_1)\phi_1 < c_1/2$ then

$$ |\sin (1 + 1/p_1)\phi_1 - \sin (1 + 1/p_2)\phi_1| \le (1/p_1 - 1/p_2)\phi_1 < c_1/2, $$

and consequently $\sin (1 + 1/p_2)\phi_1 < -c_1/2$.

If we take $t$ with $0 \le t < t_1$, then

$$ \begin{align*} \Psi_{p_2,t}(\phi_1) &= \Psi_{p_2,t}(\phi_1) - \Psi_{p_1,t_1}(\phi_1) \\ &= (1 - t_1)(\sin (1 + 1/p_2)\phi_1 - \sin (1 + 1/p_1)\phi_1) \\ &\quad + (\sin (1 - 1/p_2)\phi_1 - \sin (1 - 1/p_1)\phi_1) + (t_1 - t) \sin (1 + 1/p_2)\phi_1 \\ &\le (2 - t_1)(p_2 - p_1)\phi_1 - (t_1 - t)c_1/2. \end{align*} $$
Hence, if

$$ (2 - t_1)(p_2 - p_1)\phi_1 < (t_1 - t)c_1/2 $$

then $\Psi_{p_2,t}(\phi_1) < 0$. This implies that

$$ g(p_1) - g(p_2) = t_1 - t_2 \le 2(2 - t_1)(p_2 - p_1)\phi_1/c_1, $$

and proves continuity of $g$.

For $p = 3/2$ we can write

$$ \Psi_{3/2,t}(\phi) = \frac{\sin \phi/3}{4} \left[ (1-t)(5 - 8 \sin^2 \phi/3)^2 + 5t - 1 \right]. $$

Note that $\sqrt{5/8} < \sqrt{3}/2 = \sin \pi/3$, so, assuming that $0 \le t \le 1$, $\Psi_{3/2,t}$ attains its minimum on $[0, \pi]$ at $\phi = 3 \arcsin \sqrt{5/8}$. This yields $g(3/2) = 1/5$. ■

Now we are able to describe the domain of positive definiteness of the sequence $a_n(p, t)$ (see Fig 2). The density function for the particular case $p = 3/2, t = 1/5$ is illustrated in Fig. 1.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Domain of positive definiteness of the sequence $a_n(p, t)$</figcaption></figure>

**THEOREM 3.3.** Suppose that $p \ge 0$. Then the sequence $a_n(p, t)$ is positive definite if and only if $p \ge 1$ and

$$ (3.10) \qquad g(p) \le t \le \frac{2p}{1+p}. $$

*Proof.* Fix $p \ge 1$. By the definition of $g(p)$ the sequence $a_n(p, t)$ is positive definite for $t = g(p)$ and not positive definite for $t < g(p)$.

We have already observed that for $p \ge 1$ the sequence $a_n(p, 2p/(p+1))$ is positive definite. If $t > 2p/(p+1)$ then $n(2p - t - pt) + 2 < 0$, and consequently $a_n(p, t) < 0$ for all $n$ sufficiently large. Alternatively, we have $\Psi'_{p,t}(0) = 2p - pt - t < 0$ in this case, which implies $\Psi_{p,t}(x) < 0$ for some $x \in (0, \pi/p)$. ■
**4. Free transforms.** Throughout this section we assume that $p \ge 1$ and the sequence $a_n(p, t)$ is positive definite, i.e. $g(p) \le t \le 2p/(p+1)$. Denote by $\eta(p, t)$ the corresponding distribution, i.e. $\eta(1, 1) = \delta_1$ and $\eta(p, t) = f_{p,t}(x) dx$ on $[0, p^p(p-1)^{1-p}]$ for $p > 1$. We are going to study relations of these measures with free probability.

Recall that for a compactly supported probability measure $\mu$ on $\mathbb{R}$ with the moment generating function

$$ (4.1) \qquad M_{\mu}(z) := \sum_{n=0}^{\infty} z^n \int_{\mathbb{R}} x^n d\mu(x) = \int_{\mathbb{R}} \frac{1}{1-xz} d\mu(x), $$

the S- and R-transforms are defined by

$$ (4.2) \qquad M_{\mu}\left(\frac{z}{1+z}S_{\mu}(z)\right) = 1+z, $$

$$ (4.3) \qquad 1 + R_{\mu}(zM_{\mu}(z)) = M_{\mu}(z). $$

Moreover, we have the relation

$$ (4.4) \qquad R_{\mu}(zS_{\mu}(z)) = z. $$

The coefficients $r_n(\mu)$ in the Taylor expansion $R_{\mu}(z) = \sum_{n=1}^{\infty} r_n(\mu)z^n$ are called the *free cumulants* of $\mu$. It is known that $\mu$ is infinitely divisible with respect to the additive free convolution if and only if the sequence $\{r_{n+2}(\mu)\}_{n=0}^{\infty}$ is positive definite [28, 19].

For the distributions $\eta(p, t)$ we have

$$ M_{\eta(p,t)}(z) := \sum_{n=0}^{\infty} a_n(p,t)z^n = t\mathcal{B}_p(z) + (1-t)\mathcal{B}_p(z)^2. $$

Now we are going to compute the S-transform of $\eta(p, t)$.

**PROPOSITION 4.1.** For $p > 1$ and $g(p) \le t \le 2p/(p+1)$ we have

$$ (4.5) \qquad S_{\eta(p,t)}(w) = (2+2w)^{1-p} \frac{\sqrt{(2-t)^2+4(1-t)w+t}+t}{\sqrt{(2-t)^2+4(1-t)w+2-t}}. $$

*Proof.* From (2.2) we can derive the relation

$$ \mathcal{B}_p(z(1+z)^{-p}) = 1+z $$

(see [13]). Therefore

$$ M_{\eta(p,t)}(z(1+z)^{-p}) = t(1+z) + (1-t)(1+z)^2. $$

If we substitute

$$ t(1+z) + (1-t)(1+z)^2 = 1+w $$

then

$$ z = \frac{\sqrt{(2-t)^2 + 4(1-t)w - 2 + t}}{2(1-t)} = \frac{2w}{\sqrt{(2-t)^2 + 4(1-t)w + 2 - t}} $$
and

$$1 + z = \frac{\sqrt{(2-t)^2 + 4(1-t)w} - t}{2(1-t)} = \frac{2(1+w)}{\sqrt{(2-t)^2 + 4(1-t)w} + t},$$

which combined with (4.2) yields (4.5). ■

Now we are going to compute the R-transform of $\eta(p, t)$ for $p = 2$ and $p = 3$. We will denote $r_n(p, t) := r_n(\eta(p, t))$.

**4.1. The case $p = 2$.** The density function $f_{2,t}$ is given by (3.6), $0 \le t \le 4/3$. From (4.5) we can compute the R-transform for $p = 2$:

**PROPOSITION 4.2.** $R_{\eta(2,1)} = z/(1-z)$, and for $t \neq 1$,

$$R_{\eta(2,t)}(z) = \frac{1 - t - 2z + 3tz - z^2 + (t - 1 - z)\sqrt{1 + z(2 - 4t) + z^2}}{2(t - 1)}.$$

Moreover, $\eta(2,t)$ is infinitely divisible with respect to the additive free convolution if and only if either $t = 0$ or $1 \le t \le 4/3$.

*Proof.* First we find $R_{\eta(2,t)}(z)$ by solving the equation

$$S_{\eta(2,t)}(R_{\eta(2,t)}(z))R_{\eta(2,t)}(z) = z,$$

equivalent to (4.4), with the condition $R_{\eta(2,t)}(0) = 0$. In particular we have $R_{\eta(2,0)}(z) = 2z + z^2$, which implies that $\eta(2,0)$ is infinitely divisible with respect to the additive free convolution.

Now we can find

$$r_1(2,t) = 2 - t,$$

$$r_2(2,t) = 1 + t - t^2,$$

$$r_3(2,t) = 3t^2 - 2t^3,$$

$$r_4(2,t) = -4t^2 + 10t^3 - 5t^4.$$

Since

$$r_2(2,t)r_4(2,t) - r_3(2,t)^2 = t^2(t-1)(t-2)(t^2-2),$$

for $0 < t < 1$ the distribution $\eta(2,t)$ is not infinitely divisible with respect to the additive free convolution.

For $t \neq 1$ we have

$$1 + R_{\eta(2,t)}(z) = \frac{t - 1 - 2z + 3tz - z^2 + (t - 1 - z)\sqrt{1 + z(2 - 4t) + z^2}}{2(t - 1)}$$

and $1 + R_{\eta(2,1)}(z) = 1/(1-z)$. Then for $1 < t \le 3/2$ the function

$$\frac{1 + R_{\eta(2,t)}(1/z)}{z} = \frac{(t-1)z^2 - 2z + 3tz - 1 + (z(t-1) - 1)\sqrt{1 + z(2-4t) + z^2}}{2(t-1)z^3}$$
is the Cauchy transform of the probability distribution

$$ \frac{(1 - tx + x)\sqrt{4t(t-1) - (x - 2t + 1)^2}}{2\pi(t-1)x^3} dx $$

on the interval

$$ x \in [2t - 1 - 2\sqrt{t^2 - t}, 2t - 1 + 2\sqrt{t^2 - t}]. $$

Therefore for $1 < t \le 4/3$,

$$ (4.6) \quad r_n(2, t) = \int_{2t-1-2\sqrt{t^2-t}}^{2t-1+2\sqrt{t^2-t}} x^n \frac{(1-tx+x)\sqrt{4t(t-1)-(x-2t+1)^2}}{2\pi(t-1)x^3} dx, $$

which proves that the sequence ${r_{n+2}(2, t)}_{n=0}^{\infty}$ is positive definite. ■

REMARK. Note that for $\eta(2, 0)$ the cumulant sequence is $(2, 1, 0, 0, \dots)$, so the sequence ${r_{n+2}(2, 0)}_{n=0}^{\infty} = (1, 0, 0, \dots)$ is positive definite. Actually, $\eta(2, 0)$, given by (2.11), is a translation of the Wigner semicircle distribution $\frac{1}{2\pi}\sqrt{4-x^2} dx$, $x \in [-2, 2]$. The free additive infinite divisibility of $\eta(2, 0)$ was overlooked in [16, Corollary 7.1], where $\eta(2, 0)$ was denoted $\mu(2, 2)$.

EXAMPLE 1. Define $a_0 := 1$ and $a_n := 3^n \cdot r_n(2, 4/3)$ for $n \ge 1$, yielding the sequence

$$ 1, 2, 5, 16, 64, 304, 1632, 9552, 59520, 388720, 2632864, \dots $$

Applying (4.6) for $t = 4/3$ we obtain

$$ (4.7) \quad a_n = \int_1^9 x^n \frac{\sqrt{(x-1)(9-x)^3}}{2\pi x^3} dx. $$

Its generating function is

$$ (4.8) \quad \sum_{n=0}^{\infty} a_n z^n = 1 + R_{\eta(2,4/3)}(3z) = \frac{1 + 18z - 27z^2 + \sqrt{(1-z)(1-9z)^3}}{2}. $$

EXAMPLE 2. Now consider the binomial transform of $a_n$:

$$ b_n := \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} a_k, $$

for which the density function is that of the sequence $a_n$ translated by $-1$:

$$ (4.9) \quad b_n = \int_0^8 x^n \frac{\sqrt{x(8-x)^3}}{2\pi(x+1)^3} dx. $$
For the generating function we have

$$ 
\begin{aligned}
\sum_{n=0}^{\infty} b_n z^n &= \sum_{k=0}^{\infty} a_k (-1)^k \sum_{n=k}^{\infty} \binom{n}{k} (-z)^n \\
&= \sum_{k=0}^{\infty} a_k \frac{z^k}{(1+z)^{k+1}} = \frac{1}{1+z} (1 + R_{\eta(2,4/3)}(3z/(1+z))),
\end{aligned}
$$

so from (4.8),

$$ (4.10) \qquad \sum_{n=0}^{\infty} b_n z^n = \frac{1 + 20z - 8z^2 + \sqrt{(1-8z)^3}}{2(1+z)^3}. $$

This proves that $b_n$ coincides with A022558 of OEIS:

1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, ...,

which counts the permutations of length $n$ which avoid the pattern 1342 [5, Theorem 2].

## 4.2. The case $p=3$

**PROPOSITION 4.3.** We have

$$ (4.11) \qquad R_{\eta(3,t)}(z) = \frac{z(4-7t+4t^2-2z)-(t-1)^2+(1-2t+t^2-tz)\sqrt{1-4tz}}{2(t+z-1)^2}, $$

and the distribution $\eta(3,t)$ is infinitely divisible with respect to the additive free convolution if and only if $1/2 \le t \le 3/2$.

*Proof.* The proof is similar to that for $p=2$. First we find $R_{\eta(3,t)}$ by solving the equation

$$ S_{\eta(3,t)}(R_{\eta(3,t)}(z))R_{\eta(3,t)}(z) = z $$

with the condition $R_{\eta(3,t)}(0) = 0$. Then we find that

$$ 1 + R_{\eta(3,t)}(z) = \frac{(t-1)^2 + tz(4t-3) + (1-2t+t^2-tz)\sqrt{1-4tz}}{2(t+z-1)^2} $$

is the moment generating function for the density

$$ (4.12) \qquad \frac{(t-x(t-1)^2)\sqrt{4t-x}}{2\pi(tx-x+1)^2\sqrt{x}}, \quad x \in [0, 4t], $$

which is positive provided $1/2 \le t \le 3/2$. ■

**EXAMPLE.** The sequence $a_n = A220910(n)$:

1, 1, 3, 14, 83, 570, 4318, 35068, 299907, 2668994, 24513578, ...
counts matchings avoiding the pattern 231 (see [4] for details). Its generating function equals

$$ (4.13) \quad M(z) = \sum_{n=0}^{\infty} a_n z^n = \frac{1 + 36z + \sqrt{(1-12z)^3}}{2(1+4z)^2} = 1 + R_{\eta(3,3/2)}(2z), $$

so $a_n = 2^n \cdot r_n(3, 3/2)$ for $n \ge 1$. Therefore these numbers can be represented as moments:

$$ (4.14) \quad a_n = \int_0^{12} x^n \frac{\sqrt{(12-x)^3}}{2\pi(x+4)^2\sqrt{x}} dx. $$

Now we are going to prove a recurrence relation, which was conjectured by R. J. Mathar (see OEIS, entry A220910, Aug. 04 2013).

**PROPOSITION 4.4.** For $n \ge 2$ we have

$$ (4.15) \quad na_n = (8n-34)a_{n-1} + 24(2n-3)a_{n-2}. $$

*Proof.* One can check that the generating function satisfies the differential equation

$$ (1 - 8z - 48z^2)M'(z) + (26 - 24z)M(z) = 27. $$

The coefficient of $z^{n-1}$ on the left hand side is equal to

$$ na_n - 8(n-1)a_{n-1} - 48(n-2)a_{n-2} + 26a_{n-1} - 24a_{n-2} $$

for $n \ge 2$, which gives (4.15).

Now we will provide two formulas for $a_n = A220910(n)$.

**PROPOSITION 4.5.**

$$ (4.16) \quad a_n = \frac{1-8n}{2}(-4)^n + \binom{2n}{n} \sum_{k=0}^{n} \frac{3^{n+1}(k+1) \prod_{i=0}^{k-1} (n-i)}{8(-3)^k \prod_{i=0}^{k+1} (n-i-1/2)} $$

$$ (4.17) \quad = \frac{(-4)^n (1-8n)}{16} \left[ 8 - \sum_{k=0}^{n+1} \frac{(-3)^k}{k!} \prod_{i=0}^{k-1} (i-3/2) \right] + \binom{2n}{n} \frac{3^{n+3}}{32(n+1)}. $$

*Proof.* Setting $x = 12t$ in (4.14) and applying [20, (15.6.1)] we get

$$ (4.18) \quad a_n = \frac{9 \cdot 12^n}{2\pi} \int_0^1 \frac{t^{n-1/2}(1-t)^{3/2}}{(1+3t)^2} dt $$

$$ (4.19) \quad = \frac{27(2n)!3^n}{8n!(n+2)!} {}_2F_1(2, n+1/2; n+3|-3). $$

From [20, (15.8.2)] and the identities

$$ \frac{\Gamma(n-3/2)}{\Gamma(n+1/2)} = \frac{4}{(2n-3)(2n-1)}, \quad \frac{\Gamma(3/2-n)}{\Gamma(5/2)} = \frac{(-2)^{n+1}(2n-1)}{3(2n-1)!!}, $$
we have

$$ \begin{aligned} {}_2F_1(2, n + 1/2; n + 3 | -3) & = \frac{4(n + 2)!}{9n!(2n - 1)(2n - 3)} {}_2F_1(2, -n; 5/2 - n | -1/3) \\ & \quad + \frac{(-2)^{n+1}(n + 2)!(2n - 1)}{3^{n+3/2}(2n - 1)!!} {}_2F_1(n + 1/2, -3/2; n - 1/2 | -1/3). \end{aligned} $$

Since

$$ {}_2F_1(2, -n; 5/2 - n | z) = \sum_{k=0}^{n} (k+1)z^k \prod_{i=0}^{k-1} \frac{n-i}{n-5/2-i} $$

and

$$ {}_2F_1(n + 1/2, -3/2; n - 1/2 | z) = \frac{(2n - 2nz - 2z - 1)\sqrt{1-z}}{2n - 1} $$

(see [20, (15.4.9)]), we obtain

$$ \begin{aligned} {}_2F_1(2, n + 1/2; n + 3 | -3) = & \frac{n!(n + 2)!(8n - 1)(-4)^{n+1}}{(2n)!3^{n+3}} \\ & + \frac{4(n + 1)(n + 2)}{9(2n - 1)(2n - 3)} \sum_{k=0}^{n} \frac{k + 1}{(-3)^k} \prod_{i=0}^{k-1} \frac{n - i}{n - 5/2 - i} \end{aligned} $$

which leads to (4.16).

For the second formula we apply the identity

$$ {}_2F_1(2, b; c|z)(1-z) = (bz - z - c + 2) {}_2F_1(1, b; c|z) + c - 1 $$

(see [20, (15.5.11)]) to (4.19) and get

$$ {}_2F_1(2, n + 1/2; n + 3 | -3) = \frac{1 - 8n}{8} {}_2F_1(1, n + 1/2; n + 3 | -3) + \frac{n + 2}{4}. $$

Applying, [23, (123), p. 462]:

$$ {}_2F_1(1, b; m + 1 | z) = \frac{m!}{z^m (b-1) \dots (b-m)} \left( (1-z)^{m-b} - \sum_{k=0}^{m-1} \frac{z^k}{k!} \prod_{i=0}^{k-1} (b+i-m) \right) $$

with $b = n + 1/2$, $m = n + 2$, $z = -3$, and using the identity

$$ 4^{n+1}n!(n + 1/2 - 1) \dots (n + 1/2 - n - 2) = 3(2n)! $$

we get (4.17). ■

**Acknowledgments.** W. M. is supported by the Polish National Science Center grant No. 2012/05/B/ST1/00626.
REFERENCES

[1] N. Alexeev, F. Götze and A. Tikhomirov, *Asymptotic distribution of singular values of powers of random matrices*, Lithuanian Math. J. 50 (2010), 121–132.

[2] D. Armstrong, *Generalized noncrossing partitions and combinatorics of Coxeter groups*, Mem. Amer. Math. Soc. 202 (2009), no. 949, x + 159 pp.

[3] T. Banica, S. Belinschi, M. Capitaine and B. Collins, *Free Bessel laws*, Canad. J. Math. 63 (2011), 3–37.

[4] J. Bloom and S. Elizalde, *Pattern avoidance in matchings and partitions*, Electron. J. Combin. 20 (2013), no. 2, #5, 38 pp.

[5] M. Bona, *Exact enumeration of 1342-avoiding permutations; A close link with labeled trees and planar maps*, J. Combin. Theory Ser. A 80 (1997), 257–272.

[6] M. Bousquet-Mélou and G. Schaeffer, *Enumeration of planar constellations*, Adv. Appl. Math. 24 (2000), 337–368.

[7] P. H. Edelman, *Chain enumeration and non-crossing partitions*, Discrete Math. 31 (1980), 171–180.

[8] P. J. Forrester and D.-Z. Liu, *Raney distributions and random matrix theory*, J. Statist. Phys. 158 (2015), 1051–1082.

[9] R. L. Graham, D. E. Knuth and O. Patashnik, *Concrete Mathematics. A Foundation for Computer Science*, Addison-Wesley, New York, 1994.

[10] U. Haagerup and S. Møller, *The law of large numbers for the free multiplicative convolution*, Operator Algebra and Dynamics, Springer Proc. Math. Statist. 58, T. M. Carlsen et al. (eds.), Springer, Berlin, 2013, 157–186.

[11] D. Liu, C. Song and Z. Wang, *On explicit probability densities associated with Fuss-Catalan numbers*, Proc. Amer. Math. Soc. 139 (2011), 3735–3738.

[12] J.-G. Liu and R. Pego, *On generating functions of Hausdorff moment sequences*, Trans. Amer. Math. Soc. 368 (2016), 8499–8518.

[13] W. Młotkowski, *Fuss-Catalan numbers in noncommutative probability*, Documenta Math. 15 (2010), 939–955.

[14] W. Młotkowski, M. A. Nowak, K. A. Penson and K. Życzkowski, *Spectral density of generalized Wishart matrices and free multiplicative convolution*, Phys. Rev. E 92 (2015), 012121, 9 pp.

[15] W. Młotkowski and K. A. Penson, *The probability measure corresponding to 2-plane trees*, Probab. Math. Statist. 33 (2013), 255–264.

[16] W. Młotkowski and K. A. Penson, *Probability distributions with binomial moments*, Infinite Dimensional Anal. Quantum Probab. Related Topics 17 (2014), 1450014, 32 pp.

[17] W. Młotkowski, K. A. Penson and K. Życzkowski, *Densities of the Raney distributions*, Documenta Math. 18 (2013), 1573–1596.

[18] T. Neuschel, *Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution*, Random Matrices Theory Appl. 3 (2014), 1450003, 18 pp.

[19] A. Nica and R. Speicher, *Lectures on the Combinatorics of Free Probability*, Cambridge Univ. Press, Cambridge, 2006.

[20] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, *NIST Handbook of Mathematical Functions*, Cambridge Univ. Press, Cambridge, 2010.

[21] K. A. Penson and A. I. Solomon, *Coherent states from combinatorial sequences*, Quantum Theory and Symmetries (Kraków, 2001), World Sci., River Edge, NJ, 2002, 527–530.
[22] K. A. Penson and K. Życzkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions, Phys. Rev. E 83 (2011), 061118, 9 pp.

[23] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, *Integrals and Series*, Vol. 3: More Special Functions, Gordon and Breach, Amsterdam, 1998.

[24] J. H. Przytycki and A. S. Sikora, *Polygon dissections and Euler, Fuss, Kirkman, and Cayley numbers*, J. Combin. Theory Ser. A 92 (2000), 68-76.

[25] A. Schuetz and G. Whieldon, *Polygonal dissections and reversions of series*, Involve 9 (2016), 223-236.

[26] T. Simon, *Comparing Fréchet and positive stable laws*, Electron. J. Probab. 19 (2014), no. 16, 25 pp.

[27] N. J. A. Sloane, *The On-Line Encyclopedia of Integer Sequences*, http://oeis.org/ (2017).

[28] D. V. Voiculescu, K. J. Dykema and A. Nica, *Free Random Variables*, CRM Monograph Ser. 1, Amer. Math. Soc., 1992.

Wojciech Młotkowski

Instytut Matematyczny

Uniwersytet Wrocławski

Plac Grunwaldzki 2/4

50-384 Wrocław, Poland

E-mail: mlotkow@math.uni.wroc.pl

Karol A. Penson
Sorbonne Universités
Université Pierre et Marie Curie
Laboratoire de Physique Théorique
de la Matière Condensée
CNRS UMR 7600, Tour 13, 5ième ét.
Boîte Courrier 121, 4 place Jussieu
F-75252 Paris Cedex 05, France
E-mail: penson@lptl.jussieu.fr