# q-Pascal and q-Wronskian Matrices with Implications to  q-Appell Polynomials

Authors: Thomas Ernst
Journal: Journal of Discrete Mathematics
 (2013)

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## Abstract

We introduce aq-deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for q-Appell polynomials, and in particular for q-Bernoulli and q-Euler polynomials. Furthermore, the q-L-polynomial, anticipated by Ward, can be expressed as a sum of products of q-Bernoulli and q-Euler polynomials. The pseudo q-Appell polynomials, which are first presented in this paper, enable multiple q-analogues of the Yang and Youn formulas. The generalized q-Pascal functional matrix, the q-Wronskian vector of a function, and the vector of q-Appell polynomials together with the q-deformed ϵ matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of q-Appell numbers, improving on Al-Salam 1967. Finally, we find a q-difference equation for the q-Appell polynomial of degree n.


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## Body

## 1. Introduction

In this paper we will introduce theq-Pascal and q-Wronskian matrices in a general setting, with the aim of fruitful applications for q-Appell polynomials. These two matrices contain a certain q-difference operator, just like the q-deformed Leibniz functional matrix from [1]. By the q-Leibniz formula, products of such matrices contain a certain operator ·ϵ, just as in the previous article, but with a slightly different definition. However, since in almost all equations we compute the function value at t=0, this operator will convert to ordinary matrix multiplication by formula (6). There is a connection to the q-Pascal matrix [2] for the special case f=q-exponential function. The Appell polynomials are seldom encountered in the literature; one exception was Carlitz’ article [3], where a formula for the product of two generating functions was given. Carlitz cleverly observed that the generating function for Appell polynomials can be inverted when f(0)=1. The NWA q-addition seldom occurs in the literature, except for the author’s papers; it was first seen in an Italian paper by Nalli [4]. Corresponding to the two dual q-additions, NWA and JHC, there are two dual q-Bernoulli polynomials, with the same names; these are special cases of q-Appell polynomials, the content of Section 2. It turns out that for certain purposes we will need pseudo q-Appell polynomials, which correspond to the other basic q-addition, JHC; these polynomials were first seen in [5]. The q-Appell numbers form a certain algebraic structure, with two operations, which was first described by Al-Salam [6]. This structure will in a certain way be generalized to q-Appell polynomials. The q-Bernoulli and q-Euler polynomials are presented along with the q-Lucas polynomials; these definitions coincide with those given in [7]. In Section 3 we come to the core of the paper, which starts with the q-Pascal matrix and the q-Wronskian vector. In the beginning, we briefly pass to the shorter paper [8], to find a few q-analogues of truncated q-exponential generating function formulas. We mostly follow the ordering in [9]. Sometimes multiple q-analogues follow, and these will usually come with the pseudo q-Appell polynomials. A couple of general formulas for q-Appell polynomials are given; sometimes these are specialized to q-Bernoulli and q-Euler polynomials. In the last part, we find a q-difference equation for general q-Appell polynomials. In an appendix we compare our formulas for Bernoulli and Euler polynomials with Hansen’s table [10], where another notation is used.We now start with the definition of the umbral method from the recent book [11]. Some of the notations are well known and will be skipped.Definition 1.
Let the Gaussq-binomial coefficient be defined by
(1)(nk)q≡{n}q!{k}q!{n-k}q!,k=0,1,…,n.
Let a and b be any elements with commutative multiplication. Then the NWA q-addition is given by
(2)(a⊕qb)n≡∑k=0n‍(nk)qakbn-k,n=0,1,2,….
The JHC q-addition is the function:
(3)(a⊞qb)n≡∑k=0n‍(nk)qq(k2)bkan-k=an(-ba;q)n,n=0,1,2,….
The q-exponential function is defined by
(4)Eq(z)≡∑k=0∞‍1{k}q!zk.
Theq-derivative is defined by
(5)(Dqf)(x)≡{f(x)-f(qx)(1-q)x,ifq∈ℂ∖{1},x≠0,dfdx(x)ifq=1,dfdx(0)ifx=0.
Assume that t, y, and q are the variables. If we want to indicate the variable on which the q-difference operator is applied, we denote the operator in (5) by (Dq,tφ)(t,y). To save space, we sometimes write f′ instead of Dq,tf.Theorem 2.
The following equation relates thenth q-difference operator at zero to the nth derivative at zero for an analytic function f(z) [12],
(6)Dqnf(0)={n}q!n!f(n)(0).For convenience, we will use the following two abbreviations for functionsf(t): f(k) represents the kth q-derivative of f and fk represents the kth power of f. We will only consider functions in ℂ[[t]]. Sometimes functions can be given in other forms, we then mean the Taylor expansion of the function around zero. In the sense of Milne-Thomson [13] and Rainville [14] we write =¨ to indicate the symbolic nature.Iff(x)∈ℂ[x], the ring of polynomials with complex coefficients, then the function ϵ:ℂ[x]↦ℂ[x] is defined by
(7)ϵf(x)≡f(qx).
We make the convention that all n×n matrices are denoted by the first index n. The rows and columns are denoted by i and j. The q-Pascal matrix [2] is defined by
(8)Pn,q(x)(i,j)≡(ij)qxi-j.
The Polya-Vein matrix Hn,q [2] is given by
(9)Hn,q(i,i-1)≡{i}q,i=1,…,n-1,Hn,q(i,j)≡0,j≠i-1.


## 2.q-Appell Polynomials

The Appell polynomials have been around for more than a century, but this general notion is still not so well known. The usual definition is(10)f(t)ext=∑ν=0∞‍tνν!Φν(x),f(t)∈ℂ[[x]].
The original article by Appell [15] started with the definition dAn/dx=nAn-1. Then Appell showed that multiplication of two Appell polynomials is commutative. He then considered the inverse of An as a natural generalization of ordinary division. The q-Appell polynomials form a natural generalization of the Appell polynomials.


### 2.1. Definitions

We consider a certain value of the order (=1) for the q-Appell polynomials in [11]. The degree is only a name; it is not the degree of polynomials.Definition 3.
Theq-Appell polynomials Φν,q(x) of degree ν are defined by
(11)f(t)Eq(xt)=∑ν=0∞‍tν{ν}q!Φν,q(x),f(t)∈ℂ[[x]].
We say that Φν,q(x) is the q-Appell polynomial sequence for f(t).
We putx=0 and obtain
(12)f(t)=∑ν=0∞tν{ν}q!Φν,q,Φ0,q=1,
where Φν,q is said to have degree ν.Definition 4.
The pseudoq-Appell polynomials Φνq(x) of degree ν [5] are defined by
(13)f(t)E1/q(xt)=∑ν=0∞‍tν{ν}q!Φνq(x).
We say that Φνq(x) is the pseudo q-Appell polynomial sequence for f(t).
By puttingx=0, we have
(14)f(t)=∑ν=0∞‍tν{ν}q!Φνq,
where Φνq is called a Φq number of degree ν.
We have similarly(15)DqΦνq(x)={ν}qΦν-1,qq(qx),Φνq(x)=¨(Φq⊞qx)ν.


### 2.2.q-Analogue of Carlitz’ Product Formula (1963)

To motivate the generalizedq-Pascal functional matrix, we show that the Carlitz [3, page 247] formula for the product of generating functions for Appell polynomials has a q-analogue involving the NWA q-addition. Define the numbers {Φν,q′}ν=0∞ by the inverse of (12) (this is allowable, since f(0)=1):
(16)f(t)-1=∑ν=0∞tν{ν}q!Φν,q′,Φ0,q′=1.
Then we have the following.Theorem 5.
An inverse relation formula for theq-Appell polynomials. 									(17)xn=∑k=0n‍(nk)qΦn-k,q′Φk,q(x).Proof.
By the umbral definition ofq-Appell polynomials we have
(18)∑k=0n‍(nk)qΦn-k,q′Φk,q(x)=∑k=0n‍(nk)qΦn-k,q′∑l=0k‍(kl)qΦk-l,qxl=∑k=0n‍{n}q!Φn-k,q′{n-k}q!∑l=0n‍Φk-l,q{l}q!{k-l}q!xl=∑l=0n‍{n}q!{l}q!xl∑s+t=n-l‍Φs,q′Φt,q{s}q!{t}q!=∑l=0n‍{n}q!{l}q!xlδn-l=xn.
This completes the proof.We introduce two otherq-Appell polynomials Ξq and Ψq by
(19)g(t)Eq(xt)=∑ν=0∞‍tν{ν}q!Ξν,q(x),g(t)∈ℂ[[x]],g(t)=∑ν=0∞tν{ν}q!Ξν,q,Ξ0,q=1,h(t)Eq(xt)=∑ν=0∞‍tν{ν}q!Ψν,q(x),h(t)∈ℂ[[x]],h(t)=∑ν=0∞tν{ν}q!Ψν,q,Ψ0,q=1.
Let us consider
(20)fm(ax)gn(bx)=∑r=0m‍∑s=0n‍(mr)q(ns)qfm-rgn-sarbsxr+s.
Now put
(21)h(t)-1=∑ν=0∞tν{ν}q!Ψν,q′.
Because of  (17) we get
(22)fm(ax)gn(bx)=∑r=0m∑s=0n(mr)q(ns)qfm-rgn-sarbs∑j=0r+s(r+sj)qΨr+s-j,q′Ψj,q(x).
Introduce the notation
(23)A(j,q,m,n,a,b)≡∑r=0m∑s=0n(mr)q(ns)q(r+sj)qfm-rgn-sarbsΨr+s-j,q′,
where
(24)fm(ax)gn(bx)=∑j=0m+nA(j,q,m,n,a,b)Ψj,q(x).Theorem 6.
With the previously mentioned notation, one has the followingq-analogue of Carlitz [3, page 247]:
(25)∑m=0∞∑n=0∞A(j,q,m,n,a,b)umvn{m}q!{n}q!=f(u)g(v)∑k=0∞Ψk,q′(au⊕qbv)k+j{j}q!{k}q!.Proof.
Thus we have(26)∑m=0∞‍∑n=0∞‍A(j,q,m,n,a,b)umvn{m}q!{n}q!=∑m=0∞‍∑n=0∞‍umvn{m}q!{n}q!∑r=0m‍∑s=0n‍=(mr)q(ns)q(r+sj)qfm-rgn-sarbsΨr+s-j,q′=∑r=0∞‍∑s=0∞‍(au)r(bv)s{r}q!{s}q!,(r+sj)qΨr+s-j,q′∑m=0∞‍∑n=0∞‍fmgnumvn{m}q!{n}q!=f(u)g(v)∑r=0∞‍∑s=0∞‍(au)r(bv)s{r}q!{s}q!,(r+sj)qΨr+s-j,q′=f(u)g(v)∑k=j∞‍(kj)qΨk-j,q′∑r+s=k‍(au)r(bv)s{r}q!{s}q!=f(u)g(v)∑k=j∞‍(kj)qΨk-j,q′(au⊕qbv)k{k}q!.


### 2.3. Characterization ofq-Appell Numbers

We will now present a characterization ofq-Appell numbers compare with Al-Salam [6].Definition 7.
We denote by𝒜q the set of all q-Appell-numbers.
LetΦn,q and Ψn,q be two elements in 𝒜q. Then the operations + and ⋆ are defined as follows:
(27)(Φq+Ψq)n≡Φn,q+Ψn,q,(28)(Φq⋆Ψq)n≡∑k=0n‍(nk)qΦn-k,qΨk,q.Theorem 8.
The identity element in𝒜q with respect to ⋆ is I≡δn,0. This corresponds to f(t)=1.Proof.
Putk=n and Φq=1 in (28).Theorem 9 (Compare with [6, page 34]).
LetΦn,q, Ξn,q and Ψn,q be three elements in 𝒜q corresponding to the generating functions f(t), g(t), and h(t), respectively. Further assume that f(0)+h(0)≠0. Then
(29)Φn,q+Ψn,q∈𝒜q,Φn,q+Ψn,qhasgeneratingfunctionf(t)+h(t).
The associative law for addition reads:
(30)Φn,q+(Ξn,q+Ψn,q)=(Φn,q+Ξn,q)+Ψn,q.Proof.
This follows from the definitions.Theorem 10 (Compare with [6, page 34]).
LetΦn,q, Ξn,q, and Ψn,q be three elements in 𝒜q corresponding to the generating functions f(t), g(t), and h(t), respectively. Then
(31)Φn,q⋆Ψn,q∈𝒜q,(32)Φn,q⋆Ψn,q=Ψn,q⋆Φn,q,(33)Φn,q⋆Ψn,qhasgeneratingfunctionf(t)h(t).
The associative law for ⋆ reads:
(34)Φn,q⋆(Ξn,q⋆Ψn,q)=(Φn,q⋆Ξn,q)⋆Ψn,q.Proof.
We prove (31) and (33) at the same time. We have in the umbral sense
(35)f(t)h(t)=Eq(t(Φq⊕qΨq)).
Formulas (32) and (34) follow by the commutativity and the associativity of NWA.


### 2.4. Specializations

Definition 11.
The NWAq-Bernoulli polynomials  BNWA,ν,q(x)  of degree ν are defined by
(36)t1(Eq(t)-1)Eq(xt)=∑ν=0∞‍tνBNWA,ν,q(x){ν}q!,|t|<2π.
The NWAq-Bernoulli polynomials are also given by
(37)BNWA,ν,q(x)=¨(BNWA,q⊕qx)ν.
The pseudo-q-Bernoulli polynomials are defined by
(38)BNWA,νq(x)=¨(BNWA,q⊞qx)ν.
The JHCq-Bernoulli polynomials BJHC,ν,q(x) of degree ν are defined by
(39)t1(E1/q(t)-1)Eq(xt)=∑ν=0∞‍tνBJHC,ν,q(x){ν}q!,|t|<2π.


## 2.1. Definitions

We consider a certain value of the order (=1) for the q-Appell polynomials in [11]. The degree is only a name; it is not the degree of polynomials.Definition 3.
Theq-Appell polynomials Φν,q(x) of degree ν are defined by
(11)f(t)Eq(xt)=∑ν=0∞‍tν{ν}q!Φν,q(x),f(t)∈ℂ[[x]].
We say that Φν,q(x) is the q-Appell polynomial sequence for f(t).
We putx=0 and obtain
(12)f(t)=∑ν=0∞tν{ν}q!Φν,q,Φ0,q=1,
where Φν,q is said to have degree ν.Definition 4.
The pseudoq-Appell polynomials Φνq(x) of degree ν [5] are defined by
(13)f(t)E1/q(xt)=∑ν=0∞‍tν{ν}q!Φνq(x).
We say that Φνq(x) is the pseudo q-Appell polynomial sequence for f(t).
By puttingx=0, we have
(14)f(t)=∑ν=0∞‍tν{ν}q!Φνq,
where Φνq is called a Φq number of degree ν.
We have similarly(15)DqΦνq(x)={ν}qΦν-1,qq(qx),Φνq(x)=¨(Φq⊞qx)ν.


## 2.2.q-Analogue of Carlitz’ Product Formula (1963)

To motivate the generalizedq-Pascal functional matrix, we show that the Carlitz [3, page 247] formula for the product of generating functions for Appell polynomials has a q-analogue involving the NWA q-addition. Define the numbers {Φν,q′}ν=0∞ by the inverse of (12) (this is allowable, since f(0)=1):
(16)f(t)-1=∑ν=0∞tν{ν}q!Φν,q′,Φ0,q′=1.
Then we have the following.Theorem 5.
An inverse relation formula for theq-Appell polynomials. 									(17)xn=∑k=0n‍(nk)qΦn-k,q′Φk,q(x).Proof.
By the umbral definition ofq-Appell polynomials we have
(18)∑k=0n‍(nk)qΦn-k,q′Φk,q(x)=∑k=0n‍(nk)qΦn-k,q′∑l=0k‍(kl)qΦk-l,qxl=∑k=0n‍{n}q!Φn-k,q′{n-k}q!∑l=0n‍Φk-l,q{l}q!{k-l}q!xl=∑l=0n‍{n}q!{l}q!xl∑s+t=n-l‍Φs,q′Φt,q{s}q!{t}q!=∑l=0n‍{n}q!{l}q!xlδn-l=xn.
This completes the proof.We introduce two otherq-Appell polynomials Ξq and Ψq by
(19)g(t)Eq(xt)=∑ν=0∞‍tν{ν}q!Ξν,q(x),g(t)∈ℂ[[x]],g(t)=∑ν=0∞tν{ν}q!Ξν,q,Ξ0,q=1,h(t)Eq(xt)=∑ν=0∞‍tν{ν}q!Ψν,q(x),h(t)∈ℂ[[x]],h(t)=∑ν=0∞tν{ν}q!Ψν,q,Ψ0,q=1.
Let us consider
(20)fm(ax)gn(bx)=∑r=0m‍∑s=0n‍(mr)q(ns)qfm-rgn-sarbsxr+s.
Now put
(21)h(t)-1=∑ν=0∞tν{ν}q!Ψν,q′.
Because of  (17) we get
(22)fm(ax)gn(bx)=∑r=0m∑s=0n(mr)q(ns)qfm-rgn-sarbs∑j=0r+s(r+sj)qΨr+s-j,q′Ψj,q(x).
Introduce the notation
(23)A(j,q,m,n,a,b)≡∑r=0m∑s=0n(mr)q(ns)q(r+sj)qfm-rgn-sarbsΨr+s-j,q′,
where
(24)fm(ax)gn(bx)=∑j=0m+nA(j,q,m,n,a,b)Ψj,q(x).Theorem 6.
With the previously mentioned notation, one has the followingq-analogue of Carlitz [3, page 247]:
(25)∑m=0∞∑n=0∞A(j,q,m,n,a,b)umvn{m}q!{n}q!=f(u)g(v)∑k=0∞Ψk,q′(au⊕qbv)k+j{j}q!{k}q!.Proof.
Thus we have(26)∑m=0∞‍∑n=0∞‍A(j,q,m,n,a,b)umvn{m}q!{n}q!=∑m=0∞‍∑n=0∞‍umvn{m}q!{n}q!∑r=0m‍∑s=0n‍=(mr)q(ns)q(r+sj)qfm-rgn-sarbsΨr+s-j,q′=∑r=0∞‍∑s=0∞‍(au)r(bv)s{r}q!{s}q!,(r+sj)qΨr+s-j,q′∑m=0∞‍∑n=0∞‍fmgnumvn{m}q!{n}q!=f(u)g(v)∑r=0∞‍∑s=0∞‍(au)r(bv)s{r}q!{s}q!,(r+sj)qΨr+s-j,q′=f(u)g(v)∑k=j∞‍(kj)qΨk-j,q′∑r+s=k‍(au)r(bv)s{r}q!{s}q!=f(u)g(v)∑k=j∞‍(kj)qΨk-j,q′(au⊕qbv)k{k}q!.


## 2.3. Characterization ofq-Appell Numbers

We will now present a characterization ofq-Appell numbers compare with Al-Salam [6].Definition 7.
We denote by𝒜q the set of all q-Appell-numbers.
LetΦn,q and Ψn,q be two elements in 𝒜q. Then the operations + and ⋆ are defined as follows:
(27)(Φq+Ψq)n≡Φn,q+Ψn,q,(28)(Φq⋆Ψq)n≡∑k=0n‍(nk)qΦn-k,qΨk,q.Theorem 8.
The identity element in𝒜q with respect to ⋆ is I≡δn,0. This corresponds to f(t)=1.Proof.
Putk=n and Φq=1 in (28).Theorem 9 (Compare with [6, page 34]).
LetΦn,q, Ξn,q and Ψn,q be three elements in 𝒜q corresponding to the generating functions f(t), g(t), and h(t), respectively. Further assume that f(0)+h(0)≠0. Then
(29)Φn,q+Ψn,q∈𝒜q,Φn,q+Ψn,qhasgeneratingfunctionf(t)+h(t).
The associative law for addition reads:
(30)Φn,q+(Ξn,q+Ψn,q)=(Φn,q+Ξn,q)+Ψn,q.Proof.
This follows from the definitions.Theorem 10 (Compare with [6, page 34]).
LetΦn,q, Ξn,q, and Ψn,q be three elements in 𝒜q corresponding to the generating functions f(t), g(t), and h(t), respectively. Then
(31)Φn,q⋆Ψn,q∈𝒜q,(32)Φn,q⋆Ψn,q=Ψn,q⋆Φn,q,(33)Φn,q⋆Ψn,qhasgeneratingfunctionf(t)h(t).
The associative law for ⋆ reads:
(34)Φn,q⋆(Ξn,q⋆Ψn,q)=(Φn,q⋆Ξn,q)⋆Ψn,q.Proof.
We prove (31) and (33) at the same time. We have in the umbral sense
(35)f(t)h(t)=Eq(t(Φq⊕qΨq)).
Formulas (32) and (34) follow by the commutativity and the associativity of NWA.


## 2.4. Specializations

Definition 11.
The NWAq-Bernoulli polynomials  BNWA,ν,q(x)  of degree ν are defined by
(36)t1(Eq(t)-1)Eq(xt)=∑ν=0∞‍tνBNWA,ν,q(x){ν}q!,|t|<2π.
The NWAq-Bernoulli polynomials are also given by
(37)BNWA,ν,q(x)=¨(BNWA,q⊕qx)ν.
The pseudo-q-Bernoulli polynomials are defined by
(38)BNWA,νq(x)=¨(BNWA,q⊞qx)ν.
The JHCq-Bernoulli polynomials BJHC,ν,q(x) of degree ν are defined by
(39)t1(E1/q(t)-1)Eq(xt)=∑ν=0∞‍tνBJHC,ν,q(x){ν}q!,|t|<2π.


## 3.q-Pascal and q-Wronskian Matrices

We now come to the most important part of the paper, which is of interest even for the caseq=1, since the original paper by Yang and Youn is not very widespread.Definition 12.
The generalizedq-Pascal functional matrix is given by
(40)(𝒫n,q)[f(t,q)](i,j)≡{(ij)qDq,ti-jf(t,q)ifi≥j,i,j=0,1,2,…,n-1.0,otherwise,
In the special casef(t,q)= constant,
(41)(𝒫n,q)[f(t,q)]=f(t,q)I.Definition 13.
Theq-deformed Wronskian vector is given by
(42)(𝒲n,q)[f(t,q)](i,0)≡Dq,tif(t,q),i=0,1,…,n-1.Definition 14.
Theq-deformed Leibniz functional matrix is given by
(43)(ℒn,q)[f(t,q)](i,j)≡{Dq,ti-jf(t,q){i-j}q!ifi≥j,i,j=0,1,2,…,n-1.0,otherwise,
In order to find aq-analogue of the formula [16]
(44)(ℒn)[f(t)g(t)]=(ℒn)[f(t)](ℒn)[g(t)],
we infer that by the q-Leibniz formula [1]
(45)(ℒn,q)[f(t,q)g(t,q)]=(ℒn,q)[f(t,q)]·ϵ(ℒn,q)[g(t,q)],
where in the matrix multiplication for every term which includes Dqkf, we operate with ϵk on g. We denote this by ·ϵ. This operator can also be iterated, compare with [1].Theorem 15 (aq-analogue of [9, page 67]).
Consider(46)𝒫n,q[Eq(xt)]|t=0=Pn,q(x),
where the q-Pascal matrix is defined in (8).
The mappings𝒫n,q[·] and 𝒲n,q[·] are linear, that is,
(47)𝒫n,q[af(t)+bg(t)]=a𝒫n,q[f(t)]+b𝒫n,q[g(t)],𝒲n,q[af(t)+bg(t)]=a𝒲n,q[f(t)]+b𝒲n,q[g(t)].
One has the following two matrix multiplication formulas: the first one is aq-analogue of [8, page 232],
(48)𝒫n,q[f(t,q)]·ϵ𝒫n,q[g(t,q)]=𝒫n,q[f(t,q)g(t,q)].(49)𝒫n,q[f(t,q)]·ϵ𝒲n,q[g(t,q)]=𝒲n,q[f(t,q)g(t,q)].
The scaling properties can be stated as follows:
(50)𝒲n,q[f(at,q)]|t=0=diag[1,a,…,an-1]𝒲n,q[f(t,q)]|t=0,(51)𝒫n,q[f(at,q)]|t=0=diag[1,a,…,an-1]𝒫n,q[f(t,q)]|t=0×diag[1,a-1,…,a1-n].Proof.
The formula (49) is proved by the q-Leibniz theorem. For (50) use the chain rule.Definition 16.
Assume thatf(0)≠0. If the inverse (f(t)-1)(k) exists for k<n, we can define the q-inverse of the generalized q-Pascal functional matrix as
(52)𝒫n,q[f(t,q)]-1,q≡𝒫n,q[f(t,q)-1].Example 17 (aq-analogue of [8, pages 231–234]).
Let(53)(𝒢n,q)[x](i,j)≡{gi-j(x)(ij)qifi≥j,i,j=0,1,2,…,n-1.0,otherwise,
Put
(54)f(t,x,q)≡∑k=0n‍gk(x)tk{k}q!,
a truncated q-exponential generating function of q-binomial type. We find that
(55)𝒫n,q[f(t,x,q)]|t=0=𝒢n,q[x].
By a routine computation we obtain
(56)𝒢n,q[x]𝒢n,q[y]=𝒢n,q[x⊕qy].
The function argument x⊕qy is induced from the corresponding q-addition.
In the same way we obtain(57)𝒢n,q[x]𝒢n,1/q[y]=𝒢n,q[x⊞qy].
By the last formula we can define the inverse of 𝒢n,q[x] as
(58)𝒢n,q[x]-1≡𝒢n,1/q[-x].
We now return permanently to [9].Definition 18 (compare with [9, page 71]).
LetΦq(x) be the q-Appell polynomial corresponding to f(t) in (11). The vector of q-Appell polynomials Φ→q(x) for f(t) is given by
(59)Φ→q(x)(i)≡Φi,q(x),i=0,1,…,n-1.Theorem 19 (aq-analogue of [9, page 69]).
Letf(t) be analytic in the neighbourhood of 0, with f(0)=0. The q-difference equation in ℝn,
(60)(Dq,x)y→(x)=𝒫n,q[f(t,q)]|t=0y→(x),-∞<x<∞,
with initial value y→(0)=y→0, has the following solution:
(61)y→(x)=𝒫n,q[Eq(f(t)x)]|t=0y→0.Proof.
SinceEq(At)|t=0 is equal to the unit matrix, the solution of (60) is
(62)Φq(x)=Eq(𝒫n,q[f(t)]|t=0x)y→0=(∑k=0∞‍𝒫n,qk[f(t)])|t=0xk{k}q!.
By (47) and (48)
(63)Φq(x)=𝒫n,q(∑k=0∞‍fk(t)xk{k}q!)|t=0,
where the power fk(t) is to be interpreted as k-1 products with ·ϵ.Theorem 20 (aq-analogue of [9, page 71]).
LetΦ→(x) be a vector of length n. Then Φ→(x) is the q-Appell polynomial vector for f(t) if and only if
(64)Φ→(x)=𝒫n,q[f(t)]|t=0·ϵ𝒲n,q[Eq(xt)]|t=0=𝒲n,q[f(t)Eq(xt)]|t=0.Proof.
We know from [2, 17] that
(65)𝒫n,q[t]|t=0=Hn,q.
We thus obtain
(66)Φ→(x)=𝒫n,q[Eq(xt)]|t=0·ϵ𝒲n,q[f(t)]|t=0=𝒲n,q[f(t)Eq(xt)]|t=0.Definition 21 (Aq-analogue of [9, page 74]).
LetΦ→(x) be the q-Appell polynomial vector for f(t). Then the q-Pascal matrix for the q-Appell polynomial vector for f(t) is defined by
(67)𝒫n,q[f(t)Eq(xt)]|t=0.
Can be compared with (53).Theorem 22 (aq-analogue of [9, pages 74-75]).
LetΦν,q(x),  Ξν,q(x), and Ψν,q(x) be the q-Appell polynomials corresponding to f(t), g(t), and h(t), respectively.
Further assume thatf(t)h(t)=g(t).
Then(68)Ξν,q(y⊕qz)=∑k=0ν(νk)qΦν-k,q(y)Ψk,q(z).Proof.
By formula (49) we get
(69)𝒫n,q[f(t)Eq(yt)]·ϵ𝒲n,q[h(t)Eq(zt)]|t=0=𝒲n,q[f(t)h(t)Eq((y⊕qz)t)]|t=0.
By (11) the left-hand side of (69) is equal to(70)(Φ0,q(y)0⋯0Φ1,q(y)Φ0,q(y)⋯0Φ2,q(y){2}qΦ1,q(y)⋯0⋮⋮⋱⋮(n-10)qΦn-1,q(y)(n-11)qΦn-2,q(y)⋯(n-1n-1)qΦ0,q(y))×(Ψ0,q(z)Ψ1,q(z)⋮Ψn-1,q(z)).By (11) the right-hand side of (69) is equal to
(71)(Ξ0,q(y⊕qz)Ξ1,q(y⊕qz)⋮Ξn-1,q(y⊕qz)).
The theorem follows by equating the last rows of formulas (71) and (70).Formula (68) is almost a generalization of (28).There is a variation of this formula with JHC, which enables inverses.Theorem 23 (anotherq-analogue of [9, pages 74-75]).
LetΦν,q andΞν,q be two q-Appell polynomial sequences for f(t) and g(t), respectively. Let Ψνq be the pseudo q-Appell polynomial for h(t). Furthermore assume that f(t)h(t)=g(t). Then
(72)Ξν,q(y⊞qz)=∑k=0ν(νk)qΦν-k,q(y)Ψkq(z).Proof.
By formula (49) we get
(73)𝒫n,q[f(t)Eq(yt)]|t=0𝒲n,q[h(t)E1/q(zt)]|t=0=𝒲n,q[f(t)h(t)Eq((y⊞qz)t)]|t=0.
By (11) and (13) the left-hand side of (73) is equal to(74)(Φ0,q(y)0⋯0Φ1,q(y)Φ0,q(y)⋯0Φ2,q(y){2}qΦ1,q(y)⋯0⋮⋮⋱⋮(n-10)qΦn-1,q(y)(n-11)qΦn-2,q(y)⋯(n-1n-1)qΦ0,q(y))×(Ψ0q(z)Ψ1q(z)⋮Ψn-1q(z)).By (11) the right-hand side of (73) is equal to
(75)(Ξ0,q(y⊞qz)Ξ1,q(y⊞qz)⋮Ξn-1,q(y⊞qz)).
The theorem follows by equating the last rows of formulas (75) and (74).The following formula from [18, equation (50), page 133] for q=1 follows immediately from (68).Corollary 24 ([11, 4.242]).
When∑l=1s‍nl=n,
(76)BNWA,k,q(n)(x1⊕q⋯⊕qxs)=∑m1+⋯+ms=k‍(km1,…,ms)q∏j=1s‍BNWA,mj,q(nj)(xj),
where we assume that nj operates on xj.Yang and Youn have found a new identity between Bernoulli and Euler polynomials:(77)2nBn(y+z2)∑k=0ν‍(νk)Bk(y)En-k(z).
The q-analogue looks as follows.Corollary 25 (aq-analogue of [9, page 75 (18)]).
Consider(78)LNWA,ν,q(y⊕qz)=∑k=0ν(νk)qBNWA,ν-k,q(y)FNWA,ν,q(z).Proof.
We put inf(t)=t1/(Eq(t)-1) and h(t)=2/(Eq(t)+1) in (68). Then
(79)gn(t)=f(t)g(t)=(2t)1(Eq(t2¯q)-1).
This is obviously the same as the left hand side of the generating function for qL-numbers.Corollary 26 (aq-analogue of [9, pages 85-86]).
LetΦn,q(x) be a q-Appell polynomial and Ψnq(x) a pseudo q-Appell polynomial. Then
(80)∑k=0n‍(nk)qΦk,q(x)Ψn-kq(-x)=∑k=0n‍(nk)qΦk,qΨn-k,q.
Let Φn,q(x) and Ψn,q(x) be q-Appell polynomials. Then
(81)∑k=0n‍(nk)qΦk,q(y)Ψn-k,q(z)=∑k=0n‍(nk)qΦk,q(y⊕qz)Ψn-k,q.Proof.
Formula (80) is proved as follows. We put z=-y in (72). The RHS of (72) is then equal to the LHS of (80). We finish the proof by using formula (49). Formula (81) is proved as follows. Use (68) with f(t)→f(t)Eq(t(y⊕qz)) and h(t). Or use an equivalent umbral reasoning.Corollary 27.
One has the following special cases:(82)∑k=0n‍(nk)qBNWA,k,q(x)BNWA,n-kq(-x)=∑k=0n‍(nk)qBNWA,k,qBNWA,n-k,q,(83)∑k=0n‍(nk)qBNWA,k,q(y)BNWA,n-k,q(z)=∑k=0n‍(nk)qBNWA,k,q(y⊕qz)BNWA,n-k,q,(84)∑k=0n‍(nk)qFNWA,k,q(y)BNWA,n-k,q(z)=∑k=0n‍(nk)qFNWA,k,q(y⊕qz)BNWA,n-k,q.Corollary 28 (aq-analogue of [9, page 87]).
Consider theq-Appell polynomial Φn,q(x) and the pseudo-q-Appell polynomial Ψnq(x). Then(85)∑k=0n‍(-1)k(nk)qΦk,q(x)Ψn-kq(x)=∑k=0n‍(-1)k(nk)qΦk,qΨn-k,q.Proof.
LetΦn,q(x) be the q-Appell polynomial for f(t) and Ψnq(x) the pseudo-q-Appell polynomial for h(t). By formula (50),(86)𝒫n,q[f(t)Eq(xt)]·ϵ𝒲n,q[h(-t)E1/q(-xt)]|t=0=(Φ0,q(x)0⋯0Φ1,q(x)Φ0,q(x)⋯0Φ2,q(x){2}qΦ1,q(x)⋯0⋮⋮⋱⋮(n-10)qΦn-1,q(x)(n-11)qΦn-2,q(x)⋯(n-1n-1)qΦ0,q(x))(Ψ0q(x)-Ψ1q(x)⋮(-1)n+1Ψn-1q(x)).On the other hand, by formula (49), we obtain(87)𝒫n,q[f(t)Eq(xt)]·ϵ𝒲n,q[h(-t)E1/q(-xt)]|t=0=𝒫n,q[f(t)]·ϵ𝒲n,q[h(-t)]|t=0=(Φ0,q(0)0⋯0Φ1,q(0)Φ0,q(0)⋯0Φ2,q(0){2}qΦ1,q(0)⋯0⋮⋮⋱⋮(n-10)qΦn-1,q(0)(n-11)qΦn-2,q(0)⋯(n-1n-1)qΦ0,q(0))(Ψ0q(0)-Ψ1q(0)⋮(-1)n+1Ψn-1q(0)).The theorem follows by multiplying both sides of (85) by (-1)n and equating the last rows of formulas (86) and (87).We can easily write down specializations of (85) to (pseudo) q-Bernoulli and q-Euler polynomials. One example is
(88)∑k=0n‍(-1)k(nk)qBNWA,k,q(x)BNWA,n-kq(x)=∑k=0n‍(-1)k(nk)qBNWA,k,qBNWA,n-k,q.Theorem 29 (theq-connection constant theorem, a q-analogue of [9, page 77]).
LetΦ→q(x) and Ψ→q(x) be the q-Appell vectors corresponding to f(t) and g(t), respectively. Then
(89)Φ→q(x)=𝒫n,q[f(t)g(t)]|t=0Ψ→q(x).Proof.
By formula (49) we get
(90)Φ→q(x)=𝒲n,q[f(t)g(t)g(t)Eq(xt)]|t=0=𝒫n,q[f(t)g(t)]·ϵ𝒲n,q[g(t)Eq(xt)]|t=0.Corollary 30 (aq-analogue of [9, page 77]).
A relationship between Bernoulli and Euler polynomials(91)BNWA,n,q(x)={n}qFNWA,n-1,q(x)2+∑k=0n(nk)qBNWA,n-k,qFNWA,k,q(x).Proof.
We put inf(t)=t1/(Eq(t)-1) and g(t)=2/(Eq(t)+1) in (89). Then
(92)f(t)g(t)=t12+t1Eq(t)-1.
We find that
(93)(f(t)g(t))(k)(0)={12+BNWA,k,q,ifk=1,BNWA,k,qotherwise.
The result follows by (89).Theorem 31 (aq-analogue of [9, page 91]).
LetΦn,q(x) be the q-Appell polynomial sequence for f(t). Then Φn,q(x) satisfies the following q-difference equation:
(94)∑k=2n‍qnαk-1{k-1}q!Dqkf(xq-1)+qn(xq+α0)Dqf(xq-1)-{n+1}qf(x)+qnf(xq-1)=0,
where αk≡Dqk(-f(t)Dq(1/f(t)))|t=0.Proof.
We putf(t)=1/g(t):(95)(Φ1,q(x)Φ2,q(x)⋮Φn,q(x))=𝒲n,q[Dq(Eq(xt)g(t))]|t=0=𝒲n,q[(x-Dqg(t)g(t))Eq(xt)g(qt)]|t=0=by(50)𝒫n,q[Eq(xt)g(qt)]·ϵ𝒲n,q[x-Dqg(t)g(t)]|t=0=(Φ0,q(xq-1)0⋯0qΦ1,q(xq-1)Φ0,q(xq-1)⋯0q2Φ2,q(xq-1)q{2}qΦ1,q(xq-1)⋯0⋮⋮⋱⋮qn-1(n-10)qΦn-1,q(xq-1)qn-2(n-11)qΦn-2,q(xq-1)⋯(n-1n-1)qΦ0,q(xq-1))(x+α0α1⋮αn-1).By equating the first and last expression for the last rows of formula (95) we get (ν=0,1,…,n)
(96)Φν,q(x)=(x+α0)qν-1Φν-1,q(xq-1)+∑k=1ν-1‍(ν-1k)qαkqν-k-1Φν-k-1,q(xq-1).
Operating with Dq on this and putting ν-1=n,
(97){n+1}qΦn,q(x)=qnΦn,q(xq-1)+qn-1{n}q(qx+α0)Φn-1,q(xq-1)+∑k=1n‍(nk)qαkqn-k-1{n-k}q×Φn-k-1,q(xq-1),n=0,1,…,n,
where we have used the following identities:
(98)DqkΦn,q(xq-1)=q-k{n-k+1}k,qΦn-k,q,DqnΦn-1,q(xq-1)=0.


## 4. Conclusion

By findingq-analogues of Yang and Youn [9], we have also shown that Yang-Youn paper contains a lot of new and interesting formulas. It is our hope that this paper together with future articles will point out the interplay between special functions and linear algebra. The formulas in Corollary 28 are in the same style as Nørlund’s [18] formulas, which are proved using umbral calculus. However, our formulas (78), Corollary 30, and (91) are of a more intricate nature. This shows that the matrix approach in this paper is not just a restatement of Nørlund’s [18] umbral calculus in q-deformed form but a generalization. We also see that in order to obtain q-analogues of the Yang and Youn [9, pages 85-87] formulas, we had to use (pseudo) q-Bernoulli and q-Euler polynomials, which were first mentioned in [5].



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