Summation formula for generalized discrete $q$-Hermite II polynomials

TL;DR

This paper introduces a new family of generalized discrete q-Hermite II polynomials, establishes their relations with other q-polynomials, and derives a summation formula using generating functions.

Contribution

It presents a novel family of generalized discrete q-Hermite II polynomials and connects them with q-Laguerre and Stieltjes-Wigert polynomials, along with a new summation formula.

Findings

Explicit relations with q-Laguerre and Stieltjes-Wigert polynomials

Derived summation formula for the polynomials

Enhanced understanding of their generating functions

Abstract

In this paper, we provide a family of generalized discrete $q$-Hermite II polynomials denoted by $\tilde{h}_{n,\alpha}(x,y|q)$. An explicit relations connecting them with the $q$-Laguerre and Stieltjes-Wigert polynomials are obtained. Summation formula is derived by using different analytical means on their generating functions.

Equations

$${\mathcal{H}}_{2n}^{(\mu )}(x;q):$$

$${\mathcal{H}}_{2n+1}^{(\mu )}(x;q):$$

$$L_n^{(\alpha )}(x;q):$$

$$=$$

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

$$\sum _{k=0}^{\infty }{\left[(-1{)}^k{q}^{k(k-1)/2}\right]}^{1+s-r}\frac{(q^{-n};q{)}_k(a_2;q{)}_k\cdots (a_r;q{)}_k}{(b_1;q{)}_k(b_2;q{)}_k\cdots (b_s;q{)}_k}\frac{x^k}{(q;q{)}_k}$$

$${\int }_{-\infty }^{\infty }{\mathcal{H}}_n^{(\mu )}(x;q){\mathcal{H}}_m^{(\mu )}(x;q)\omega (x)dx=\pi q^{-n/2}(q^{1/2};q^{1/2}{)}_n(q^{1/2};q{)}_{1/2}{\delta }_{nm}$$

$${\tilde{h}}_{2n,\alpha }(x;q)$$

$${\tilde{h}}_{2n+1,\alpha }(x;q)$$

$${\int }_{-\infty }^{+\infty }{\tilde{h}}_{n,\alpha }(x;q){\tilde{h}}_{m,\alpha }(x;q){\omega }_{\alpha }(x;q)|x{|}^{2\alpha +1}{d}_{q}x$$

$$=\frac{2q^{-n^2}(1-q)(-q,-q,q^2;q^2{)}_{\infty }}{(-q^{-2\alpha -1},-q^{2\alpha +3},q^{2\alpha +2};q^2{)}_{\infty }}\frac{(q;q{)}_n^2}{(q;q{)}_{n,\alpha }}{\delta }_{n,m}$$

$$(a;q{)}_0=1;(a;q{)}_n=\prod _{k=0}^{n-1}(1-a{q}^k),n=1,2,\cdots ;(a;q{)}_{\infty }=\prod _{k=0}^{\infty }(1-a{q}^k)$$

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

$$\displaystyle\big{(}x\oplus_{q}y\big{)}^{n}:$$

$$=$$

$$\big{(}x\ominus_{q}y\big{)}^{n}:=\big{(}x\oplus_{q}(-y)\big{)}^{n}.$$

$$[n+1{]}_{q,\alpha }!=[n+1+{\theta }_n(2\alpha +1){]}_q[n{]}_{q,\alpha }!$$

$$(q;q{)}_{n+1,\alpha }=(1-q)[n+1+{\theta }_n(2\alpha +1){]}_q(q;q{)}_{n,\alpha },$$

$$\displaystyle\theta_{n}=\left\{\begin{array}[]{ll}1&\mbox{ if n even }\\ 0&\mbox{ if n odd}.\end{array}\right.$$

$$(q;q{)}_{n,-1/2}=(q;q{)}_n,[n{]}_{q,-1/2}!=(1-q{)}^n(q;q{)}_n.$$

$$(q;q{)}_{2n,\alpha }=(q^2;q^2{)}_n(q^{2\alpha +2};q^2{)}_n,$$

$$(q;q{)}_{2n+1,\alpha }=(q^2;q^2{)}_n(q^{2\alpha +2};q^2{)}_{n+1}.$$

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

$$e_q(x)=\sum _{k=0}^{\infty }\frac{x^k}{(q;q{)}_k}=\frac{1}{(x;q{)}_{\infty }},|x|<1.$$

$${\tilde{E}}_{q^m,\alpha }(x):=\sum _{k=0}^{\infty }\frac{q^{mk(k-1)/2}{x}^k}{(q^m;q^m{)}_{k,\alpha }},$$

$${\tilde{e}}_{q^m,\alpha }(x):=\sum _{k=0}^{\infty }\frac{x^k}{(q^m;q^m{)}_{k,\alpha }},|x|<1.$$

$${\tilde{E}}_{q,\alpha }(x)=E_q(x),{\tilde{e}}_{q,\alpha }(x)=e_q(x).$$

$${\tilde{e}}_{q^2,-\frac{1}{2}}(x){\tilde{E}}_{q^2,-\frac{1}{2}}(y)={\tilde{e}}_{q^2,-\frac{1}{2}}(x{\oplus }_{q^2}y),$$

$${\tilde{e}}_{q,-\frac{1}{2}}(x){\tilde{E}}_{q^2,-\frac{1}{2}}(-y)={\tilde{e}}_q(x{\ominus }_{q,q^2}y),{\tilde{e}}_{q^2,-\frac{1}{2}}(x){\tilde{E}}_{q^2,-\frac{1}{2}}(-x)=1,$$

$$(a{\ominus }_{q,q^2}b{)}^n:=n{!}_q\sum _{k=0}^n\frac{(-1{)}^k{q}^{k(k-1)}}{(n-k){!}_{q}k{!}_{q^2}}{a}^{n-k}{b}^k,(a{\ominus }_{q,q^2}b{)}^0:=1.$$

$${\tilde{h}}_{n,\alpha }(x,y|q):=(q;q{)}_n\sum _{k=0}^{\lfloor n/2\rfloor }\frac{(-1{)}^k{q}^{-2nk+k(2k+1)}{x}^{n-2k}{y}^k}{(q;q{)}_{n-2k,\alpha }(q^2;q^2{)}_k}$$

$${\tilde{h}}_{n,\alpha }(x,0|q):=\frac{(q;q{)}_n}{(q;q{)}_{n,\alpha }}{x}^n.$$

Full text

1. Introduction

In their paper, Àlvarez-Nodarse et al [2], have introduced a $q$-extension of the discrete $q$-Hermite II polynomials as:

[TABLE]

where $\mu>-1/2$, $L_{n}^{(\alpha)}(x;q)$ are the $q$-Laguerre polynomials given by

[TABLE]

with $(a;q)_{0}=1$, $\displaystyle(a;q)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),\,n=1,2,\cdots$, the $q$-shifted factorial, and

[TABLE]

[TABLE]

the usual generalized basic or q-hypergeometric function of degree $n$ in the variable $x$ (see Slater [10, Chap. 3], Srivastava and Karlsson [11, p.347, Eq. (272)] for details). For $\mu=0$ in (1), the polynomials $\mathcal{H}_{n}^{(0)}(x;q)$ correspond to the discrete $q$-Hermite II polynomials [1, 8], i.e., $\mathcal{H}_{n}^{(0)}(x;q^{2})=q^{n(n-1)}\tilde{h}_{n}(x;q).$ They show that the polynomials $\mathcal{H}_{n}^{(\mu)}(x;q)$ satisfy the orthogonality relation

[TABLE]

on the whole real line $\mathbb{R}$ with respect to the positive weight function $\omega(x)=1/(-x^{2};q)_{\infty}$. A detailed discussion of the properties of the polynomials $\mathcal{H}_{n}^{(\mu)}(x;q)$ can be found in [2].

Recently, Saley Jazmat et al [7], introduced a novel extension of discrete $q$-Hermite II polynomials by using new $q$-operators. This extension is defined as:

[TABLE]

For $\alpha=-1/2$ in (1), the polynomials $\tilde{h}_{n,-\frac{1}{2}}(x;q)$ correspond to the discrete $q$-Hermite II polynomials, i.e., $\tilde{h}_{n,-\frac{1}{2}}(x;q)=\tilde{h}_{n}(x;q).$ The generalized discrete $q$-Hermite II polynomials (1) satisfy the orthogonality relation

[TABLE]

[TABLE]

on the whole real line $\mathbb{R}$ with respect to the positive weight function $\omega_{\alpha}(x)=1/(-q^{-2\alpha-1}\,x^{2};q^{2})_{\infty}$. A detailed discussion of the properties of the polynomials $\tilde{h}_{n,\alpha}(x;q)$ can be found in [7].

Srivastava and Jain [12, 6], investigated multilinear generating functions for $q$-Hermite, $q$-Laguerre polynomials and other special functions. Relevant connections of these multilinear generating functions with various known results for the classical or $q$-Hermite polynomials are also indicated. They also proved many combinatorial $q$-series identities by applying the theory of $q$-hypergeometric functions (see [6], for more details).

Motivated by Saley Jazmat’s work [7], our interest in this paper is to introduce new family of “generalized discrete $q$-Hermite II polynomials (in short gdq-H2P) $\tilde{h}_{n,\alpha}(x,y|q)$” which is an extension of the generalized discrete $q$-Hermite II polynomials $\tilde{h}_{n,\alpha}(x;q)$ and investigate summation formulae.

The paper is organized as follows. In Section 2, we recall notations to be used in the sequel. In Section 3, we define a gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ and investigate several properties. In Section 4, we derive summation and inversion formulae for gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$. In Section 5, concluding remarks are given.

2. Notations and Preliminaries

For the convenience of the reader, we provide in this section a summary of the mathematical notations and definitions used in this paper. We refer to the general references [4, 8] and [7] for the definitions and notations. Throughout this paper, we assume that $0<q<1,\,\alpha>-1$.

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

[TABLE]

and the $q$-number is defined by:

[TABLE]

Let $x$ and $y$ be two real or complex numbers, the Hahn [5] $q$-addition $\oplus_{q}$ of $x$ and $y$ is given by:

[TABLE]

while the $q$-subtraction $\ominus_{q}$ is given by

[TABLE]

The generalized $q$-shifted factorials [7] are defined by the recursion relations

[TABLE]

and

[TABLE]

where

[TABLE]

Remark that, for $\alpha=-1/2$, we have

[TABLE]

We denote

[TABLE]

and

[TABLE]

The two Euler’s $q$-analogues of the exponential functions are given by [4]

[TABLE]

and

[TABLE]

For $m\geq 1$ and by means of the generalized $q$-shifted factorials, we define two generalized $q$-exponential functions as follows

[TABLE]

and

[TABLE]

Remark that, for $m=1$ and $\alpha=-\frac{1}{2}$, we have:

[TABLE]

For $m=2,$ the following elementary result is useful in the sequel to establish the summation formulae for gdq-H2P:

[TABLE]

[TABLE]

where

[TABLE]

3. Generalized discrete $q$-Hermite II polynomials

In this section, we introduce a sequence of gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$. Several properties related to these polynomials are derived.

Definition 3.1**.**

For $x,\,y\in\,\mathbb{R},$ the gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$ are defined by:

[TABLE]

and

[TABLE]

Remark that,

1. (1)

for $\displaystyle\;y=1$, we get

[TABLE]

where $\tilde{h}_{n,\alpha}(x;q)$ is the generalized discrete $q$-Hermite II polynomial [7]; 2. (2)

for $\displaystyle\,\alpha=-1/2$ and $y=1$, we have

[TABLE]

where $\tilde{h}_{n}(x;q)$ is the discrete $q$-Hermite II polynomial [1, 8]. 3. (3)

Indeed since

[TABLE]

one readily verifies that

[TABLE]

where $h_{n}^{\alpha+\frac{1}{2}}(x)$ is the Rosenblum’s generalized Hermite polynomial [9].

Lemma 3.1**.**

The following recursion relation for gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$ holds true.

[TABLE]

[TABLE]

Proof.

To prove the assertion (3.7), we consider separately even and odd cases of the expression

[TABLE]

For $n$ even, we have:

[TABLE]

The right-hand side of the last relation can be written as

[TABLE]

[TABLE]

In the same way,

[TABLE]

[TABLE]

Change k to $k-1$ in (3.10), one obtains

[TABLE]

Then combining (3.9) and (3.11), we have

[TABLE]

[TABLE]

[TABLE]

After simplification, it is equal to

[TABLE]

[TABLE]

The last expression can be written as

[TABLE]

Summarizing the above calculations in (3.12)-(3.13), we get the assertion (3.7) for $n$ even. In the odd case, the proof follows the same steps as the even case. ∎

Theorem 3.1**.**

We have:

[TABLE]

and

[TABLE]

where $S_{n}(x;q)$ are the Stieltjes-Wigert polynomials [8].

In order to prove Theorem 3.1, we need the following Lemma.

Lemma 3.2**.**

For $\alpha>-1$, the sequence of gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$ can be written in terms of $q$-Laguerre polynomials $L_{n}^{(\alpha)}(x;q)$ as

[TABLE]

and

[TABLE]

In order to prove Lemma 3.2, we need the following Proposition.

Proposition 3.1**.**

For $\alpha>-1$, the sequence of gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$ can be written in terms of basic hypergeometric functions as

[TABLE]

Proof.

In fact, for $n$ even, and by using

[TABLE]

the gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ defined in (3.1) can be rewritten as

[TABLE]

From the formula [8, p.9, Eq. (0.2.12)]

[TABLE]

we have for $a=q^{2}$ and $q^{2\alpha+2}$,

[TABLE]

After simplification, the last equation reads

[TABLE]

In the odd case, the proof follows the same steps as the even case. ∎

Now, we are in position to prove Lemma 3.1.

Proof.

(of Lemma 3.1) For $n$ even, the relation (3.18) becomes:

[TABLE]

By taking $a^{-1}=q^{-2\alpha-2}$ and $z=-q^{2n+1}\,x^{2}y^{-1}$ and the formula [8, p.17, Eq. (0.6.17)]

[TABLE]

we have

[TABLE]

[TABLE]

By using (1.4), the relation (3.25) can be written as

[TABLE]

The assertion (3.16) of Lemma 3.1 follows by summarizing the above calculations in (3.23)-(3.26).

In the odd case, the proof follows the same steps as the even case. ∎

Proof.

(of Theorem 3.2) By taking the limit $\alpha\to+\infty$ in the assertions (3.16) and (3.17) of Lemma 3.1, respectively, we get the assertions (3.14) and (3.15) of Theorem 3.2. ∎

4. Connection formulae for the generalized discrete $q$-Hermite II polynomials $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$

We begin this section with the following theorem:

Theorem 4.1**.**

The sequence of gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$, which is defined by the relation (3.1), satisfies the connection formula

[TABLE]

To prove Theorem 4.1, we need the following Lemma.

Lemma 4.1**.**

The following generating function for gdq-H2P $\{\tilde{h}_{n,\alpha}(x,y|q)\}_{n=0}^{\infty}$ holds true.

[TABLE]

Proof.

Let us consider the function

[TABLE]

By replacing in (4.3) gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ by its explicit expression (3.1) we obtain

[TABLE]

The right-hand side of (4.4) also reads

[TABLE]

Next, changing $n-2k$ by $r,\;r=0,1,\cdots$, the last relation becomes

[TABLE]

Hence,

[TABLE]

∎

Now, we are in position to prove Theorem 4.1.

Proof.

(of Theorem 4.1) Replacing $t$ by $u\oplus_{q}t$ in (4.2), we find the following generating function

[TABLE]

which by using (2.20), becomes

[TABLE]

Replacing $y$ by $\omega$ and (4.9), respectively, in (4.8), we get

[TABLE]

[TABLE]

By using (2.20), the last relation reads

[TABLE]

[TABLE]

According to (2.15), the right-hand side of (4.11) can be written as

[TABLE]

Let us substitute $n+2r=k\leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ r\leq\lfloor\,k/2\,\rfloor$ in (4.12), then we have:

[TABLE]

Next, replacing (4.13) in (4.11), we obtain

[TABLE]

[TABLE]

Finally, on equating the coefficients of like powers of $(u\oplus_{q}t)^{n}/(q;q)_{n}$ in (4.14), we get the desired identity. ∎

We have the following special cases of Theorem 4.1 of particular interest.

Corollary 4.1**.**

Letting:

• (i)

$y=0$* in the assertion (4.1) of Theorem 4.1, we get the definition of gdq-H2P (3.1), i.e.,*

[TABLE]

• (ii)

$\omega=0$* in the assertion (4.1) of Theorem 4.1, and using (3.2), we get the inversion formula for gdq-H2P*

[TABLE]

• iii)

For $y=1$, the summation formulae (4.1) can be expressed in terms of generalized discrete $q$-Hermite II polynomials $\tilde{h}_{n,\alpha}(x;q)$. Also, the summation formulae (4.1) can be written in terms of discrete $q$-Hermite II polynomials $\tilde{h}_{n}(x;q)$ by choosing $y=1$ and $\alpha=-1/2.$

5. Concluding remarks

In the previous sections, we have introduced gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ and derived several properties. Also, we have derived implicit summation formula for gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ by using different analytical means on their generating function. This process can be extended to summation formulae for more generalized forms of $q$-Hermite polynomials. This study is still in progress.

We note that the generating function of even and odd gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ are given by

[TABLE]

and

[TABLE]

where $J_{\nu}^{(2)}(z;q)$ is the $q$-analogue of the Bessel function [8].

Indeed, it is well known that from (4.2), the generating function of gdq-H2P $\tilde{h}_{n,\alpha}(x,y|q)$ is given by

[TABLE]

which on separating the power in the right-hand side into their even and odd terms by using the elementary identity

[TABLE]

becomes

[TABLE]

[TABLE]

Now replacing $t$ by $i\,t$ in (5.3) and equating the real and imaginary parts of the resultant equation, we get the generating function of even and odd gdq-H2P $h_{n,\alpha}(x,y|q)$ as

[TABLE]

and

[TABLE]

where the generalized $q$-Cosine and $q$-Sine are defined as:

[TABLE]

By using (2.12) and (2.13), respectively, the relations (5.6) and (5.7) can be expressed in terms of basic hypergeometric functions as

[TABLE]

The $q$-analogue of the Bessel function is defined [8, p.20, Eq.(0.7.14)] by

[TABLE]

from which the generating functions of (5.10) and (5.13) follow.