Some homogeneous $q$-difference operators and the associated generalized Hahn polynomials

TL;DR

This paper introduces new homogeneous $q$-difference operators and uses them to analyze generalized Hahn polynomials, deriving various $q$-identities and formulas.

Contribution

It constructs specific homogeneous $q$-shift and $q$-difference operators and applies them to represent and study generalized Hahn polynomials.

Findings

Derived generating functions for generalized Hahn polynomials

Established Mehler's and Roger's formulas in the $q$-context

Presented new $q$-identities and extended generating functions

Abstract

In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; \theta_{xy})$. We then apply these operators in order to represent and investigate generalized Cauchy and a general form of Hahn polynomials. We derive some $q$-identities such as: generating functions, extended generating functions, Mehler's formula and Roger's formula for these $q$-polynomials.

Equations

$$(a;q{)}_0:=1,(a;q{)}_n:=\prod _{k=0}^{n-1}(1-a{q}^k),n\in \mathbb{N}$$

$$(a;q{)}_{\infty }:=\prod _{k=0}^{\infty }(1-a{q}^k).$$

$$[n{]}_q:=\frac{1-q^n}{1-q},[n{]}_q!:=\prod _{k=1}^n\frac{1-q^k}{1-q},[0{]}_q!:=1.$$

$${n\text{atopwithdelims}[]k}_q:$$

$${}_{r}\Phi_{s}\left(\begin{array}[]{c}a_{1},a_{2},\ldots,a_{r}\\ b_{1},b_{2},\ldots,b_{s}\end{array}\Big{|}q;x\right)=\sum_{n=0}^{\infty}\Big{[}(-1)^{n}q^{({}^{n}_{2})}\Big{]}^{1+s-r}\,\frac{(a_{1},a_{2},\ldots,a_{r};q)_{n}}{(b_{1},b_{2},\ldots,b_{s};q)_{n}}\frac{x^{n}}{(q;q)_{n}}.$$

$$p_n(x,y):=(x-y)(x-qy)\cdots (x-q^{n-1}y)=(y/x;q{)}_n{x}^n$$

$$\sum _{k=0}^{\infty }{p}_n(x,y)\frac{t^n}{(q;q{)}_k}=\frac{(yt;q{)}_{\infty }}{(xt;q{)}_{\infty }},$$

$$p_n(x,y)=(-1{)}^n{q}^{({}_2^n)}{p}_n(y,q^{1-n}x),$$

$$p_{n-k}(x,q^{1-n}y)=(-1{)}^{n-k}{q}^{({}_2^k)-({}_2^n)}{p}_{n-k}(y,q^{k}x).$$

$$\sum_{k=0}^{\infty}\frac{(a;q)_{k}}{(q;q)_{k}}x^{k}={}_{1}\Phi_{0}\left(\begin{array}[]{c}a\\ -\end{array}\Big{|}q;x\right)=\frac{(ax;q)_{\infty}}{(x;q)_{\infty}}.$$

$$\sum _{k=0}^{\infty }\frac{x^k}{(q;q{)}_k}=\frac{1}{(x;q{)}_{\infty }}$$

$$\sum _{k=0}^{\infty }\frac{(-1{)}^k{q}^{({}_2^k)}{x}^k}{(q;q{)}_k}=(x;q{)}_{\infty }.$$

$$R(b{D}_q)=\sum _{k=0}^{\infty }\frac{(-1{)}^k{q}^{({}_2^k)}}{(q;q{)}_k}{\left(b{D}_q\right)}^k$$

$${D}_{q}\big{\{}f(a)\big{\}}=\frac{f(a)-f(qa)}{a}.$$

$$R(y{D}_q)(x^n)=p_n(x,y).$$

$$R(yD_{q})\Bigg{\{}\frac{1}{(xt;q)_{\infty}}\Bigg{\}}=\frac{(yt;q)_{\infty}}{(xt;q)_{\infty}},$$

$$R(yD_{q})\Bigg{\{}\frac{1}{(xt,xs;q)_{\infty}}\Bigg{\}}=\frac{(ys;q)_{\infty}}{(xt,xs;q)_{\infty}}{}_{1}\Phi_{1}\left(\begin{array}[]{c}xs\\ ys\end{array}\Big{|}q;yt\right),$$

$$R(yD_{q})\Bigg{\{}\frac{(xv;q)_{\infty}}{(xt,xs;q)_{\infty}}\Bigg{\}}=\frac{(ys;q)_{\infty}}{(xs;q)_{\infty}}{}_{2}\Phi_{1}\left(\begin{array}[]{c}v/t,y/x\\ ys\end{array}\Big{|}q;xt\right).$$

$$R(yD_{q})\Bigg{\{}\frac{1}{(xt,xs;q)_{\infty}}\Bigg{\}}=\frac{(ys;q)_{\infty}}{(xs;q)_{\infty}}{}_{2}\Phi_{1}\left(\begin{array}[]{c}y/x,0\\ ys\end{array}\Big{|}q;xt\right).$$

$${}_{2}\Phi_{1}\left(\begin{array}[]{c}y/x,0\\ ys\end{array}\Big{|}q;xt\right)=\frac{1}{(xt;q)_{\infty}}{}_{1}\Phi_{1}\left(\begin{array}[]{c}xs\\ ys\end{array}\Big{|}q;yt\right).$$

$$\tilde{E}(a,b;D_q)=\sum _{k=0}^{\infty }\frac{(-1{)}^k{q}^{({}_2^k)}(a;q{)}_k}{(q;q{)}_k}{\left(b{D}_q\right)}^k.$$

$$\tilde{L}(a,b;{\theta }_{xy})=\sum _{k=0}^{\infty }\frac{q^{({}_2^k)}(a;q{)}_k}{(q;q{)}_k}{\left(b{\theta }_{xy}\right)}^k,$$

$$\theta_{xy}\Big{\{}f(x,y)\Big{\}}=\frac{f(q^{-1}x,y)-f(x,qy)}{q^{-1}x-y}$$

$$\theta_{xy}^{k}\,p_{n}(y,x)=(-1)^{k}\frac{(q;q)_{n}}{(q;q)_{n-k}}p_{n-k}(y,x)\mbox{ and }\theta_{xy}^{k}\Bigg{\{}\frac{(xt;q)_{\infty}}{(yt;q)_{\infty}}\Bigg{\}}=(-t)^{k}\frac{(xt;q)_{\infty}}{(yt;q)_{\infty}}.$$

$$\tilde{L}(0,b;{\theta }_{xy})=L(b{\theta }_{xy}).$$

$$D_{q}^{n}\left\{f(x)g(x)\right\}=\sum_{k=0}^{n}\mbox{$\biggl{[}\begin{array}[]{c}n\\ k\end{array}\biggr{]}_{\!{q}}$}q^{k(k-n)}D_{q}^{k}\left\{f(x)\right\}D_{q}^{n-k}\left\{g(q^{k}x)\right\}.$$

$$D_q^n(x^k)=(q^{k-n+1};q{)}_n{x}^{k-n},D_q^n(xt;q{)}_{\infty }=q^{({}_2^n)}{t}^n\frac{(xt;q{)}_{\infty }}{(xt;q{)}_n}$$

$$D_{q}^{n}{}_{1}\Phi_{1}\left(\begin{array}[]{c}a\\ 0\end{array}\Big{|}q;bx\right)=(-1)^{n}b^{n}(a;q)_{n}q^{({}^{n}_{2})}{}_{1}\Phi_{1}\left(\begin{array}[]{c}aq^{n}\\ 0\end{array}\Big{|}q;bxq^{n}\right).$$

$$\Big{(}q^{n+m-k+1};q\Big{)}_{k}=\sum_{j=0}^{k}\mbox{$\biggl{[}\begin{array}[]{c}k\\ j\end{array}\biggr{]}_{\!{q}}$}q^{j(j-k+m)}(q^{n-j+1};q)_{j}(q^{m-k+j+1};q)_{k-j}$$

$$D_{q}^{n}\left\{\frac{{}_{1}\Phi_{1}\left(\begin{array}[]{c}a\\ 0\end{array}\Big{|}q;bsq^{k}\right)}{(xsq^{k};q)_{\infty}}\right\}$$

Full text

1. Introduction

We adopt the common conventions and notations on $q$-series. For the convenience of the reader, we provide a summary of the mathematical notations and definitions to be used in this paper. We refer to the general references (see [9]) for the definitions and notations. Throughout this paper, we assume that $0<q\leq 1$.

For complex numbers $a$, the $q$-shifted factorials are defined by:

[TABLE]

and for large $n$, we have

[TABLE]

The $q$-numbers and $q$-factorials are defined as follows:

[TABLE]

The $q$-binomial coefficients are given by

[TABLE]

The basic or q-hypergeometric function in the variable $x$ (see Slater [14, Chap. 3], Srivastava and Karlsson [16, p.347, Eq. (272)] for details) is defined as [9]:

[TABLE]

Basic or $q$-hypergeometric series and various associated families of $q$-polynomials are useful in a wide variety of fields including, for example, the theory of partitions, number theory, combinatorial analysis, finite vector spaces, Lie theory, particle physics, non-linear electric circuit theory, mechanical engineering, theory of heat conduction, quantum mechanics, cosmology, and statistics (see [16, pp. 346-351] and the references cited therein).

We will be mainly concerned with the Cauchy polynomials as given below [5]

[TABLE]

with the generating function [3]

[TABLE]

where [3]

[TABLE]

and

[TABLE]

which naturally arise in the $q$-umbral calculus [2], Goldman and Rota [10], Ihrig and Ismail [8], Johnson [7] and Roman [10]. The generating function (1.7) is also the homogeneous version of the Cauchy identity or the $q$-binomial theorem [5]

[TABLE]

Putting $a=0$, the relation (1.10) becomes Euler’s identity [5]

[TABLE]

and its inverse relation [5]

[TABLE]

Recently, Saad and Sukhi [13] have introduced the following q-exponential operator $R(b{D}_{q})$ as follows

[TABLE]

where the $q$-differential operator, or the $q$-derivative, acting on the variable a, is defined by [4, 12]

[TABLE]

Evidently,

[TABLE]

Suppose that the operator $R(yD_{q})$ acts on the variable $x$, then we have the following [13, 15]:

[TABLE]

[TABLE]

[TABLE]

Srivastava and Abdlhusein [15], showed that by setting $v=0$ in (1.18), it becomes:

[TABLE]

Comparing the identity (1.19) and the identity (1.17), we get the following transformation

[TABLE]

Motivated by Saad and Sukhi [13], Srivastava and Abdlhusein [15] works, our interest is to introduce new homogeneous $q$-difference operators $\widetilde{E}(a,b;D_{q})$ and $\widetilde{T}(a,b;\theta_{xy})$. The first homogeneous q-difference operator $\widetilde{E}(a,b;D_{q})$ is defined by

[TABLE]

Compared with $R(yD_{q})$, the homogeneous q-difference operator (1.21) involves two parameters. Clearly, the operator $R(yD_{q})$ can be considered as a special case of the operator (1.21) for $a=0$.

Second, we introduce another homogeneous q-difference operator $\widetilde{L}(a,b;\theta_{xy})$ by

[TABLE]

where [4, 12]

[TABLE]

acting on functions of suitable variables $x$ and $y$ [15, 12]

[TABLE]

The homogeneous q-difference operator (1.22) involves two parameters. It can be considered as a generalization of the homogeneous $q$-difference operator ${L}(b\theta_{xy})$ introduced by Saad and Sukhi [12]. For $a=0$, we have:

[TABLE]

The definition (1.22) is motivated by the natural question of extending generating function, Mehler’s formula and Roger’s-type formula for a general form for the Hahn polynomials $h_{n}(x,y,a,b|q)$. The operators (1.21) and (1.22) turn out to be suitable for dealing with a generalized Cauchy polynomials $p_{n}(x,y,a)$ and the generalized Hahn polynomials $h_{n}(x,y,a,b|q)$. They are then applied in order to represent and investigate some $q$-identities such as: generating functions, extended generating functions, Mehler’s formula and Roger’s-type formula for $p_{n}(x,y,a)$ and $h_{n}(x,y,a,b|q)$ polynomials.

2. Generalized Cauchy polynomials $p_{n}(x,y,a)$

In this section, we introduce a generalized Cauchy polynomials $p_{n}(x,y,a)$. We then represent this polynomials $p_{n}(x,y,a)$ by means of the homogeneous $q$-difference operator $\widetilde{E}(a,b;D_{q})$ and derive their generating function. We also use the operational formula for $p_{n}(x,y,a)$ to establish an extended generating function, Mehler’s formula and Rogers formula for the Cauchy polynomials $p_{n}(x,y,a)$.

Let us start by the the following Leibniz rule [11]

[TABLE]

For $f(x)=x^{k},\,f(x)=(xt;q)_{\infty}$ and \displaystyle f(x)={}_{1}\Phi_{1}\left(\begin{array}[]{c}a\\ 0\end{array}\Big{|}q;bx\right), respectively, we have the following identities:

[TABLE]

and

[TABLE]

Suppose that the operator $D_{q}$ acts on the variable $s$. If we set $f(s)=s^{n},\,g(s)=s^{m}$ and \displaystyle f(s)={}_{1}\Phi_{1}\left(\begin{array}[]{c}a\\ 0\end{array}\Big{|}q;bsq^{k}\right),\,\,g(s)=\frac{1}{(xsq^{k};q)_{\infty}}, respectively, and making used (2.2) and (2.1), we get the following identities to be used in the sequel:

[TABLE]

and

[TABLE]

[TABLE]

Definition 2.1**.**

The homogeneous $q$-difference operator $\widetilde{E}(a,b;D_{q})$ is defined by

[TABLE]

Remark that, for $a=0,$ we have:

[TABLE]

Proposition 2.1**.**

We suppose that the operator ${D}_{q}$ acts on the variable $x$. We have:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Setting $k=0$ in the assertion (2.15) and making used \displaystyle{}_{3}\Phi_{2}\left(\begin{array}[]{c}1,aq^{n},xt\\ 0,0\end{array}\Big{|}q;\frac{y}{x}q^{n}\right)=1, we get (2.13).

Theorem 2.1**.**

We suppose that the operator ${D}_{q}$ acts on the variable $s$. We have:

[TABLE]

[TABLE]

Proof.

[TABLE]

[TABLE]

By using (2.1), the r.h.s. of (2.17) becomes

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[TABLE]

Changing $n-k$ by $n$ in the last relation, it becomes

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By using (2.5), the last relation takes the form

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[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Summarizing the above calculations, we get the assertion (2.1). ∎

Definition 2.2**.**

In terms of $q$-shifted factorial, we define a generalized Cauchy polynomials $p_{n}(x,y,a)$ by

[TABLE]

and

[TABLE]

The following operational formula holds true.

[TABLE]

Indeed,

[TABLE]

Theorem 2.2**.**

*(Generating function for $p_{n}(x,y,a)$)

The following generating function holds true.*

[TABLE]

where $|xt|<1.$

Proof.

We suppose that the operator ${D}_{q}$ acts on the variable $x$. Then,

[TABLE]

which evidently completes the proof of assertion (2.26). ∎

Theorem 2.3**.**

*(Extended generating function for $p_{n}(x,y,a)$)

We have:*

[TABLE]

[TABLE]

where $|xt|<1.$

Proof.

We suppose that the operator ${D}_{q}$ acts on the variable $s$. Then, we have:

[TABLE]

The proof of assertion (2.32) is achieved by using the relation (2.15). ∎

Setting $k=0$ in Theorem 2.3, we get the generating function (2.26).

Theorem 2.4**.**

(Roger’s-Type formula for $p_{n}(x,y,a)$) We have:

[TABLE]

[TABLE]

where $max\{|xt|,|xs|\}<1.$

Proof.

We suppose that the operator ${D}_{q}$ acts on the variable $x$. We have:

[TABLE]

[TABLE]

The proof is achieved by using (2.14). ∎

Now, we aim to present an operator approach to Mehler’s formula for the generalized Cauchy polynomials $p_{n}(x,y,a)$.

Theorem 2.5**.**

(Mehler’s formula for $p_{n}(x,y,a)$) We have:

[TABLE]

Proof.

[TABLE]

[TABLE]

∎

For $y=0$ and $t=y$, the assertion (2.1) is reduced to the Mehler’s formula for $p_{n}(x,y,a)$ (2.35).

3. Homogeneous $q$-difference operator $\widetilde{L}(a,b;\theta_{xy})$ and generalized Hahn polynomials

In this section, we define a generalized Hahn polynomials $h_{n}(x,y,a,b|q)$ in terms of homogeneous $q$-shift operator $\widetilde{L}(a,b;\theta_{xy})$. An extended generating function, Mehler’s formula and Rogers formula for the generalized Hahn polynomials $h_{n}(x,y,a,b|q)$ are given.

Let us recall the definition of the usual operator

We now define the homogeneous Cauchy q-shift operator as follows.

Definition 3.1**.**

The homogeneous $q$-difference operator $\widetilde{L}(a,b;\theta_{xy})$ is defined by

[TABLE]

By means of $q$-hypergeometric series, the operator (3.1) can be written as:

[TABLE]

Theorem 3.1**.**

The following operational formula holds true:

[TABLE]

Proof.

Let $f_{q}(x,y;a,b;t)$ be the function defined as

[TABLE]

Using (1.24) and (3.1), respectively, we have

[TABLE]

which completes the proof of assertion (3.3). ∎

Definition 3.2**.**

In terms of the q-shifted factorial and the Cauchy polynomials $p_{n}(x,y)$, we write

[TABLE]

Remark that, for $a=0$ and $b=z$, the $q$-polynomials $h_{n}(x,y,a,b|q)$ are the well known trivariate $q$-polynomials $F_{n}(x,y,z;q)$ investigated by Mohammed (see [1] for more details), i.e.

[TABLE]

The $q$-polynomials $h_{n}(a,b,x,y|q)$ is a general form of Hahn polynomials $\psi_{n}^{(a)}(x|q)$ according when we are setting $a=0,\,b=1$ and $y=ax$. Also, if we let $a=0,y=ax$ and $b=y$, we get the second Hahn polynomials $\psi_{n}^{(a)}(x,y|q)$. Hence we have the following special substitutions:

[TABLE]

The polynomials (3.8) can be represented by the homogeneous $q$-difference operator (3.1) as follows:

[TABLE]

Indeed,

[TABLE]

Theorem 3.2**.**

*(Generating function for $h_{n}(x,y,a,b|q)$)

The following generating function holds for the $q$-polynomials $h_{n}(a,b,x,y|q)$:*

[TABLE]

Proof.

We observe that:

[TABLE]

[TABLE]

[TABLE]

which evidently completes the proof of assertion (3.13). ∎

Theorem 3.3**.**

*(Extended generating function for $h_{n}(x,y,a,b|q)$)

We have:*

[TABLE]

Proof.

We observe that:

[TABLE]

which evidently completes the proof of assertion (3.13). ∎

Setting $k=0$ in (3.14), we get the generating function (3.13) for the $q$-polynomials $h_{n}(x,y,a,b|q)$.

Theorem 3.4**.**

*(Roger’s-Type formula for $h_{n}(x,y,a,b|q)$)

The following Rogers-Type formula holds for the $q$-polynomials $h_{n}(x,y,a,b|q)$:*

[TABLE]

[TABLE]

Proof.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The proof of (3.19) of Theorem 3.4 will be completed when we replace $y$ by $x$ and $x$ by $y$, respectively, in (1.20). ∎

Theorem 3.5**.**

(Mehler’s formula for $h_{n}(x,y,a,b|q)$) We have:

[TABLE]

[TABLE]

Proof.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎