Two-dimensional $(p,q)$-heat polynomials of Gould--Hopper type

TL;DR

This paper introduces a new class of two-variable holomorphic polynomials extending Gould--Hopper polynomials, exploring their properties, identities, differential equations, and connections to hypergeometric functions.

Contribution

It presents the first comprehensive study of two-dimensional $(p,q)$-heat polynomials of Gould--Hopper type, including operational, generating, and recurrence relations.

Findings

Derived new generating functions and recurrence relations.

Established multiplication, addition, and Nielsen type formulas.

Connected these polynomials to hypergeometric functions and differential equations.

Abstract

We introduce a new class of holomorphic polynomials extending the classical Gould--Hopper to two complex variables. The considered polynomials include the $1$-D and $2$-D holomorphic and polyanalytic It\^o--Hermite polynomials as particular cases. We emphasize studying their operational representation, various generating functions, and recurrence relations. We also establish some special identities including multiplication and addition formulas of Runge type, as well as the Nielson type formulas. Higher-order partial differential equations are analyzed and the connection to Gould-Hopper polynomials and hypergeometric functions are investigated.

Equations

$$H_{n,m}(z,\bar{z}):=(-1{)}^{m+n}{e}^{|z{|}^2}{\partial }_{\bar{z}}^n{\partial }_z^m\left(e^{-|z{|}^2}\right)=n!m!\sum _{j=0}^{n\wedge m}\frac{(-1{)}^j}{j!}\frac{z^{n-j}{\bar{z}}^{m-j}}{(n-j)!(m-j)!},$$

$$H_n^{(p)}(z|\gamma )=n!\sum _{k=0}^{\lfloor \frac{n}{p}\rfloor }\frac{{\gamma }^k}{k!}\frac{z^{n-pk}}{(n-pk)!}.$$

$$H_{n,m}^{(p,q)}(z,w|\gamma )=n!m!\sum _{k=0}^{\lfloor \frac{n}{p}\rfloor \wedge \lfloor \frac{m}{q}\rfloor }\frac{{\gamma }^k}{k!}\frac{z^{n-pk}}{(n-pk)!}\frac{w^{m-qk}}{(m-qk)!}$$

$$c_{p,q}\frac{{\partial }^{p+q}}{\partial z^p\partial w^q}u(z,w;t)=\frac{\partial }{\partial t}u(z,w;t).$$

$$H_{n,m}^{(p,q)}(0,0|\gamma )=\frac{n!}{\lfloor \frac{n}{p}\rfloor !}{\gamma }^{\lfloor \frac{n}{p}\rfloor }{\delta }_{\lfloor \frac{n}{p}\rfloor ,\lfloor \frac{m}{q}\rfloor }$$

$$H_{n,m}^{(p,q)}(z,w|\gamma )=e^{\gamma {\partial }_z^p{\partial }_w^q}\{z^n{w}^m\}.$$

$$H_{n,m}^{(p,q)}(z,w|\gamma )=H_{m,n}^{(q,p)}(w,z|\gamma ).$$

$${{}_{r}F_{s}}\left(\begin{array}[]{c}a_{1},a_{2},\cdots,a_{r}\\ c_{1},c_{2},\cdots,c_{s}\end{array}\bigg{|}x\right):=\sum_{k=0}^{\infty}\frac{(a_{1})_{k}(a_{2})_{k}\cdots(a_{r})_{k}}{(c_{1})_{k}(c_{2})_{k}\cdots(c_{s})_{k}}\frac{x^{k}}{k!}.$$

$$H_{n,m}^{(p,q)}(z,w|\gamma)=z^{n}w^{m}{{}_{p+q}F_{0}}\left(\begin{array}[]{c}\frac{-n}{p},\frac{1-n}{p},\cdots,\frac{p-1-n}{p},\frac{-m}{q},\frac{1-m}{q},\cdots,\frac{q-1-m}{q}\\ \_\end{array}\bigg{|}\frac{(-p)^{p}(-q)^{q}\gamma}{z^{p}w^{q}}\right).$$

$$H_{n,m}^{(p,q)}(z,w|\gamma )=n!m!z^n{w}^m\sum _{k=0}^{\lfloor \frac{n}{p}\rfloor \wedge \lfloor \frac{m}{q}\rfloor }\frac{{\left(\gamma /z^p{w}^q\right)}^k}{k!(n-pk)!(m-qk)!}.$$

$$(n-pk)!=\frac{(-1{)}^{pk}n!}{(-p{)}^{pk}{\prod }_{j=1}^p{\left(\frac{j-1-n}{p}\right)}_k}.$$

$$H_{n,m}^{(p,q)}(z,w|\gamma )=z^n{w}^m\sum _{k=0}^{\lfloor \frac{n}{p}\rfloor \wedge \lfloor \frac{m}{q}\rfloor }\prod _{j=1}^p{\left(\frac{j-1-n}{p}\right)}_k\prod _{j=1}^q{\left(\frac{j-1-m}{q}\right)}_k\frac{{\left(\frac{(-1{)}^{p+q}{p}^p{q}^q\gamma }{z^p{w}^q}\right)}^k}{k!}.$$

$$\displaystyle{}_{2}F_{0}\left(\begin{array}[]{c}-n,-m\\ \_\end{array}\bigg{|}\frac{-1}{z}\right)=\frac{z^{-(n\wedge m)}(n\vee m)!}{(|n-m|)!}{}_{1}F_{1}\left(\begin{array}[]{c}-(n\wedge m)\\ |n-m|+1\end{array}\bigg{|}z\right).$$

$$\displaystyle{}_{1}F_{1}\left(\begin{array}[]{c}a\\ b\end{array}\bigg{|}z\right)$$

$$\displaystyle\qquad+\frac{\Gamma(b)}{\Gamma(a)}z^{a-c}e^{z}{}_{2}F_{0}\left(\begin{array}[]{c}b-a,1-a\\ \_\end{array}\bigg{|}\frac{1}{z}\right)$$

$$\sum _{n=0}^{\infty }{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{u^n}{n!}=H_m^{(q)}(w|u^p\gamma )e^{zu},$$

$$\sum _{m=0}^{\infty }{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{v^m}{m!}=H_n^{(p)}(z|v^q\gamma )e^{wv}$$

$$\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{u^n{v}^m}{n!m!}=e^{zu+wv+\gamma u^p{v}^q}.$$

$$\sum _{n=0}^{\infty }{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{u^n}{n!}=e^{\gamma {\partial }_z^p{\partial }_w^q}\{e^{zu}{w}^m\}=e^{\gamma u^p{\partial }_w^q}\{w^m\}{e}^{zu}=H_m^{(q)}(w|u^p\gamma )e^{zu}.$$

$$R_{\gamma }^{p,q}(z,w|u,v)$$

$$\sum _{m=0}^{\infty }{H}_m^{(q)}(w|\gamma )\frac{v^m}{m!}=e^{wv+\gamma v^q}.$$

$$a^n{b}^m{H}_{n,m}^{(p,q)}(z,w|\gamma )=H_{n,m}^{(p,q)}(az,bw|\gamma a^p{b}^q).$$

$$\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{a}^n{b}^m{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{u^n}{n!}\frac{v^m}{m!}=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{H}_{n,m}^{(p,q)}(az,bw|\gamma a^p{b}^q)\frac{u^n}{n!}\frac{v^m}{m!}.$$

$$\underset{t\to 0}{\lim }{\left(\frac{t}{2}\right)}^n{H}_n\left(\frac{z}{t}\right)=z^n\text{mbox}and\underset{t\to 0}{\lim }{t}^{m+n}{H}_{m,n}\left(\frac{z}{t},\frac{w}{t}\right)=z^m{w}^n$$

$$\lim_{t\to 0}t^{n+m}H_{n,m}^{(p,q)}\left(\frac{z}{t},\frac{w}{t}\Big{|}\gamma\right)=z^{n}w^{m}.$$

$$\lim_{t\to 0}t^{n+m}H_{n,m}^{(p,q)}\left(\frac{z}{t},\frac{w}{t}\Big{|}\gamma\right)=\lim_{t\to 0}H_{n,m}^{(p,q)}(z,w|t^{p+q}\gamma)=H_{n,m}^{(p,q)}(z,w|0)==0.$$

$$S_{a,b}^{p,q,\gamma }(z,w|u,v)=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }(a{)}_n(b{)}_m{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{u^n{v}^m}{n!m!}$$

$$G_{j,k}^{p,q,\gamma }(z,w|u,v):=\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }(n{)}_j(m{)}_k{H}_{n,m}^{(p,q)}(z,w|\gamma )\frac{u^n{v}^m}{n!m!}.$$

$$P_k^n(z)=\sum _{j=0}^{k-1}(-1{)}^{(k-j)}(-n{)}_{(k-j)}\left(\genfrac{}{}{0ex}{}{k}{j}\right)z^j.$$

$$G_{j,k}^{p,q,\gamma }(z,w|u,v)=uvzw{e}^{zu+wv+\gamma u^p{v}^q}\left((uz{)}^{j-1}+P_{j-1}^j(uz)\right)\left((vw{)}^{k-1}+P_{k-1}^k(vw)\right)$$

$$S_{a,b}^{p,q,\gamma }(z,w|u,v)$$

$$\displaystyle\times{}_{p+q}F_{0}\left(\begin{array}[]{c}\frac{a}{p},\frac{a+1}{p},\cdots,\frac{a+p-1}{p},\frac{b}{q},\frac{b+1}{q},\cdots,\frac{b+q-1}{q}\\ \_\end{array}\bigg{|}\frac{p^{p}q^{q}{\gamma}uv}{(1-uz)^{p}(1-vw)^{q}}\right).$$

Full text

1. Introduction

Different generalizations of the classical Hermite polynomials $H_{n}(x)$ to multivariate setting have been widely studied in the literature [14, 17, 18, 23, 24, 16] and have interesting applications in many branches of mathematics, physics and engineering. In fact, the tensor product $H_{m}(x)H_{n}(y)$ as well as the holomorphic Hermite polynomials $H_{n}(z)$ are specific $2D$ generalization of $H_{n}(x)$ to the complex plane $\mathbb{C}$. The first class is an orthogonal basis of $L^{2}(\mathbb{R}^{2};e^{-x^{2}-y^{2}}dxdy)$, while the functions $e^{-z^{2}/2}H_{m}(z)$ form an orthogonal basis of a Bargmann-Fock like space [30, 2]. For their analytic and combinatoric properties one refer to [24, 17]. The polyanalytic analog are the Itô–Hermite (or complex Hermite) polynomials defined by

[TABLE]

where $n\wedge m=min\{n,m\}$, for being solutions of the iterated Cauchy–Riemann equation $2^{n+1}\partial_{\bar{z}}^{n+1}f=(\partial_{x}+i\partial_{y})^{n+1}f=0$. These polynomials, introduced by Itô [19] in the context of the complex Markov process, are basic tools in the nonlinear analysis of traveling wave tube amplifiers [1]. More specifically, they appear in the calculation of the effects of non-linearity on broadband radio frequencies in communication systems. For their properties and applications one can refer to [7, 8, 17, 6, 9]. Their holomorphic counterparts $H_{n,m}(z,w)$ and $H_{n,m}(z,w,x)$ were introduced and studied recently in [10, 17] and [12], respectively. Another class of generalized Hermite polynomials is that we refer to as Gould–Hopper [11, Eq. (6.2), p. 58] defined by

[TABLE]

They enter in the study of Novikov–Veselov equation [3]. The specific ones $H_{n}^{(2p)}(x|(-1)^{p+1}t)$ are solutions of the higher-order heat equation associated to $(-1)^{q+1}{\partial^{2q}}/{\partial x^{2q}}$ (see e.g. [13, 11, 21]).

In the present paper, we introduce and study the basic properties of the natural extension of $H_{n}^{(p)}(z|\gamma)$ in (1.2) to two complex variables

[TABLE]

which contains all the classes mentioned earlier and gives rise to new ones. More precisely, we have

1. (1)

$H_{n,0}^{(2,0)}(2z,1|-1)=H_{n}(z)$, the holomorphic Hermite polynomials. 2. (2)

$H_{n,0}^{(p,0)}(z,1|\gamma)=H_{n}^{(p)}(z|\gamma)$, the Gould-Hopper polynomials. We also have $H_{n,m}^{(p,0)}(z,w,|\gamma)=H_{n,m}^{(0,p)}(z,w,|\gamma)=w^{m}H_{n}^{(p)}(z|\gamma).$ 3. (3)

$H_{n,m}^{(1,1)}(z,w|-1)=H_{n,m}(z,w)$ and $H_{n,m}^{(1,1)}(z,\overline{z}|-1)=H_{n,m}(z,\overline{z})$, the $2$-D holomorphic Hermite polynomials and their restriction to the non-analytic surface $w=\bar{z}$, respectively. 4. (4)

$H_{n,m}^{(1,1)}(z,\overline{z}|\gamma)=H_{n,m}(z,w,\gamma)$, the Hermite polynomials considered in [12].

Meaningful and significant application of the considered polynomials is the representation solutions of the partial differential equation in the $(z,w)$-plane

[TABLE]

with initial data of analytic. We call them here two-dimensional complex $(p,q)$-heat polynomials of Gould–Hopper type. Other motivations of considering $H_{n,m}^{(p,q)}(x,y|\gamma)$ are eventual applications in quantum mechanics, combinatorics or applied mathematics. Mainly, they can be used in evaluating transition matrix elements, studying the root dynamics of the $\sigma$-flows associated to (1.4) in a similar way as done in [29] or also in computing the higher-order moments of a given distribution.

Even if this extension is natural and some of the algebraic properties seem to be derived in a similar way, there is no evidence to suggest exact formulas or examine their analytical properties including their orthogonality, description of associated functional spaces and integral transforms. Namely, we provide a complete unified description of basic properties of $H_{n,m}^{(p,q)}(x,y|\gamma)$. More precisely, we are concerned with their operational and hypergeometric representations (Section 2), generating functions (Section 3), addition formulas of Runge type (Section 4), multiplication formulas (Section 5), recurrence relations (Section 6), Nielson type formulas (Section 7), and the connection to Gould–Hopper polynomials (Section 8). We also discuss some higher order partial differential equations (Section 9).

2. Operational and hypergeometric representations

The polynomials $H_{n,m}^{(p,q)}(z,w|\gamma)$ we deal with are the two-dimensional complex of Gould–Hopper type given through (1.3) with the convention that $H_{0,0}^{(0,0)}(z,w|\gamma)=e^{\gamma}$ and $\left\lfloor\frac{j}{k}\right\rfloor=+\infty$ when $k=0$. Obviously, they generalize the real, Gould–Hopper, as well as the $1$-D and $2$-D holomorphic, and Ito–Hermite polyanalytic complex polynomials. The monomials $z^{n}w^{m}$ can be recovered by taking $\gamma=0$ or $n<p$ and $m<q$, $H_{n,m}^{(p,q)}(z,w|\gamma)=z^{n}w^{m}$, while for $n=p$ and $m=q$, we have $H_{p,q}^{(p,q)}(z,w|0)=z^{p}w^{q}+{\gamma}p!q!.$ Notice also that the value at $z=w=0$ gives rise to

[TABLE]

when $p\mid n$ and $q\mid m$. Otherwise, we have $H_{n,m}^{(p,q)}(0,w|\gamma)=0$ if $p\nmid n$ and $H_{n,m}^{(p,q)}(z,0|\gamma)=0$ if $q\nmid m$. The case $p=n=0$, gives rise to $H_{0,qm}^{(0,q)}(0,0|\gamma)={(qm)!}\gamma^{m}/{m!}$ and $H_{0,qm+j}^{(0,q)}(0,0|\gamma)=0$ when $j=1,\dots,q-1$ so that one recoups the well-known identity for the real Hermite polynomials $H_{2n}(0)=(-1)^{n}{(2n)!}/{n!}$ and $H_{2n+1}(0)=0$.

The following operational formula can be considered as an equivalent definition which offers a broader direction, by generalizing $e^{\partial_{z}}$ to several variables and to higher order in the exponent.

Proposition 2.1**.**

We have

[TABLE]

Moreover, we have the symmetry property

[TABLE]

This follows making use of the fact that $\partial_{z}^{pk}\{z^{n}\}=\frac{n!z^{n-pk}}{(n-pk)!}$ for $k\leq\left\lfloor\frac{n}{p}\right\rfloor$ and vanishes otherwise.

The next result gives the hypergeometric representation of $H_{n,m}^{(p,q)}(z,w|\gamma)$ in terms of the ${{}_{r}F_{s}}$ hypergeometric function

[TABLE]

Proposition 2.2**.**

We have

[TABLE]

Proof.

Starting from (1.3), we can rewrite the explicit expression of $H_{n,m}^{(p,q)}$ in the following form

[TABLE]

Appealing the identity [22, Eq. (21), p. 21] as good as the Gauss multiplication theorem [22, Eq. (26), p. 23], we can rewrite the involved factorial $(n-pk)!$ as

[TABLE]

Therefore, we obtain

[TABLE]

This is exactly the desired result (2.3). ∎

Subsequently, we can express the classical classes of Hermite polynomials described in the introductory section in terms of the confluent hypergeometric function ${{}_{1}F_{1}}$. By means of the following transformation formula

[TABLE]

where $n\vee m=max(n,m)$, which can be handled by applying

[TABLE]

with $a=-n,b=m-n+1$ when $n\leq m$ and by next by the symmetry property to get the case $n\geq m$. Thus, we retrieve the ${}_{1}F_{1}$ representations [25, Eq. (1.2)] for $H_{n}(z)$ and [15, Eq. (2.3)] for $H_{n,m}^{(1,1)}(z,\bar{z})$.

3. Generating functions.

In this section, we derive generating functions for the polynomials $H_{n,m}^{(p,q)}$.

Proposition 3.1**.**

We have

[TABLE]

Proof.

We only need to establish (3.1) since the second one is its symmetry. By (2.1) we have

[TABLE]

Now, let denote the left hand-side of (3.3) by $R^{p,q}_{\gamma}(z,w|u,v)$. Hence, by means of (3.1), it follows that

[TABLE]

Then, the result in (3.3) follows using the generating function of the Gould–Hopper polynomials [4, p.72]

[TABLE]

∎

Remark 3.2**.**

For $p=q=1$, we retrieve from (3.2) the partial generating functions for the complex Hermite polynomials given through [8, Proposition 3.4], while (3.3) generalizes the corresponding formula for the Hermite polynomials described in the introductory section.

As immediate consequence of the generating function given through (3.3) is the following identity needed to prove some of our later results.

Corollary 3.3**.**

We have

[TABLE]

Proof.

Equation (3.5) follows from the observation $e^{auz+bvw+{\gamma}(au)^{p}(bv)^{q}}=e^{auz+bvw+({\gamma}a^{p}b^{q})u^{p}v^{q}}$ which combined with (3.3) yields

[TABLE]

∎

The next consequence is an extension to the $(p,q)$ complex Hermite polynomials of the limit identities [30, 10]

[TABLE]

for the holomorphic Hermite polynomials $H_{n}$ and $H_{n,m}$.

Corollary 3.4**.**

For $p+q\geq 0$, we have

[TABLE]

Proof.

By means of (3.5) with $a=b=t$ and $z:=\frac{z}{t},w:=\frac{w}{t}$, and tending $t$ to zero, we get

[TABLE]

∎

We conclude this section by establishing he closed expression of the generating functions

[TABLE]

and

[TABLE]

We set

[TABLE]

Theorem 3.5**.**

We have

[TABLE]

and

[TABLE]

Proof.

Using the definition (1.3) of $H_{n,m}^{(p,q)}$, we can rewrite $G^{p,q,\gamma}_{j,k}(z,w|u,v)$ as

[TABLE]

The second equality results from the use of the generating function [27, Theorem 2.1],

[TABLE]

Now starting from (1.3) and using making the identity [22, Eq. (5), p. 101] combined with the formula [22, Eq. (20), p. 22], we obtain

[TABLE]

Applying the Gauss multiplication theorem [22, Eq. (26), p. 23] and the binomial theorem ${{}_{1}F_{0}}(a|t)=(1-t)^{-a}$, we get

[TABLE]

which gives the right-hand side of (3.9). ∎

Remark 3.6**.**

The identity (3.8) can be seen as a special generalization of (3.3). Notice also that for$p=q=1,w=z$, and $\gamma=-1$, we retrieve [17, Eq. (4.15)]. When $p=2,m=q=0,z=2x$, and $\gamma=-1$, we get [26, p. 190]

[TABLE]

and

4. Runge formulas

In this section, we establish a Runge type formula for $H_{n,m}^{(p,q)}$. It extends the known Runge type formulas for the real Hermite polynomials [28] and the Itô–Hermite polynomials [8].

Proposition 4.1**.**

We have

[TABLE]

Proof.

We consider

[TABLE]

Using (3.3) and rewriting $e^{(z+z^{\prime})u+(w+w^{\prime})v+({\gamma}+{\gamma}^{\prime})u^{p}v^{q}}$ as $e^{zu+wv+{\gamma}u^{p}v^{q}}e^{z^{\prime}u+w^{\prime}v+{\gamma}^{\prime}u^{p}v^{q}}$, we obtain

[TABLE]

Equating the coefficients of $u^{n}v^{m}$ in the last set of equations, we obtain (4.1). ∎

Remark 4.2**.**

The special case of $\gamma=-\gamma^{\prime}$ in (4.1) leads to the following identity

[TABLE]

Remark 4.3**.**

The special case $z=z^{\prime}:=\frac{z}{2}$, $w=w^{\prime}:=\frac{w}{2}$ and $\gamma=\gamma^{\prime}=\frac{\gamma}{2}$, of (4.1) combined with (3.5) gives rise to the identity

[TABLE]

As immediate consequence of Theorem 4.1, we assert the following.

Corollary 4.4**.**

*We have *

[TABLE]

Proof.

Equation (4.2) follows from the case $\gamma=\gamma^{\prime}$ of Theorem 4.1 and equation (3.5) with $a=2^{{1}/{2p}}$ and $b=2^{{1}/{2q}}$. ∎

Remark 4.5**.**

For $p=q=1$ and $w=\bar{z}$, we recover the Runge formula for the Ito–Hermite polynomials in [8, p.9, Eq 3.22] While for $p=2$ and $m=q=0$, we recover the one for the real Hermite polynomials [28].

5. Multiplication formulas

We begin with the following multiplication formula needed to prove a recursion relation with respect to parameters $p$ and $q$.

Proposition 5.1**.**

We have the identity

[TABLE]

Proof.

From

[TABLE]

Then (5.1) follows by equating the coefficients of $u^{n}v^{m}$ on both sides of the last equation. ∎

Another multiplication formula is the following.

Proposition 5.2**.**

We have

[TABLE]

Proof.

Applying (3.5), we get

[TABLE]

By (5.1), we obtain

[TABLE]

which reduces to the right-hand side of (5.2). ∎

Remark 5.3**.**

The Gould-Hopper polynomials satisfy

[TABLE]

*We note also that for $p=2$, $q=0$, $w=1$, $c=1$ and $\gamma=-1$, we recover the multiplication formula for real Hermite polynomials $H_{n}$ in [16, Eq. (4.6.33)], while for $p=q=1$, $w=\bar{z}$, $\gamma=-1$, $b=a$ and $c=1$, we find the multiplication formula for polyanalytic polynomials $H_{n,m}$ proved in [17, Eq. (4.13)]. *

6. Recurrence formulas

In this section, we derive recurrence relations for the polynomials $H_{n,m}^{(p,q)}$. To this end, we begin by giving the action of the derivative operators with respect to $z$, $w$ and $\gamma$.

Proposition 6.1**.**

The partial derivatives of $H_{n,m}^{(p,q)}(z,w|\gamma)$ are given by

[TABLE]

Proof.

We use the operational formula (2.1) combined with the fact that $e^{\gamma\partial_{z}^{p}\partial_{w}^{q}}$ and $\partial_{z}$ commute to get (6.1). Indeed, $\partial_{z}H_{n,m}^{(p,q)}(z,w|\gamma)=e^{\gamma\partial_{z}^{p}\partial_{w}^{q}}\partial_{z}\{z^{n}w^{m}\}=ne^{\gamma\partial_{z}^{p}\partial_{w}^{q}}\{z^{n-1}w^{m}\}.$ We obtain (6.2) by the symmetry 2.2. While the third partial derivative (6.3) is obtained by noticing that $\partial_{\gamma}e^{\gamma\partial_{z}^{p}\partial_{w}^{q}}=\partial_{z}^{p}\partial_{w}^{q}e^{\gamma\partial_{z}^{p}\partial_{w}^{q}}$. ∎

Remark 6.2**.**

By mathematical induction, one can establish the following formula

[TABLE]

when $j\leq n$ and $k\leq m$. The left hand-side in (6.4) vanishes otherwise. Accordingly, we deduce

[TABLE]

if $k\leq\left\lfloor\frac{n}{p}\right\rfloor\wedge\left\lfloor\frac{m}{q}\right\rfloor$, and vanishes otherwise.

Thanks to the previous proposition, we can assert the following.

Proposition 6.3**.**

We have

[TABLE]

Subsequently, the following operational formula

[TABLE]

holds true.

Proof.

By writing down the Taylor series of the polynomials $H_{n,m}^{(p,q)}(z,w|\gamma+h)$, seen as function in the third variable, and next using (6.5), we get

[TABLE]

Then, by taking $h=-\gamma$ and using the fact that $H_{n,m}^{(p,q)}(z,w|0)=z^{n}w^{m}$, we arrive at (6.6).

The proof of (6.7) follows from operation calculus, starting from the right hand-side and making appeal of Proposition 5.2 with $a=b=1$ and $c=0$. Indeed, we have

[TABLE]

∎

The first recursion relation in this section is the following.

Proposition 6.4**.**

The polynomials $H_{n,m}^{(p,q)}$ obey the recursion relations

[TABLE]

Proof.

Starting from the generating function in (3.3) and replacing there $n$ by $n+p-1$ and $m$ by $m+q$, we get

[TABLE]

But, since $\partial_{u}e^{zu+wv+{\gamma}u^{p}v^{q}}=(z+p{\gamma}u^{p-1}v^{q})e^{zu+wv+{\gamma}u^{p}v^{q}},$ we obtain

[TABLE]

On the other hand, because of $H_{n+1,m}^{(p,q)}(z,w|\gamma)=\partial_{u}^{n}\partial_{v}^{m}(\partial_{u}e^{zu+wv+{\gamma}u^{p}v^{q}})\vline_{u=v=0},$ we conclude that

[TABLE]

This completes our check of (6.8). For the proof of (6.9), notice first that

[TABLE]

Therefore,

[TABLE]

∎

Remark 6.5**.**

The recursion relations

[TABLE]

follow from the previous ones by the symmetry identity (2.2).

In virtue of the previous recursion formulas, the polynomials $H_{n,m}^{(p,q)}$ can be rewritten, according to the values of $p$ and $q$, in terms of some creation operators with the monomials $z^{n}$ and $w^{m}$ as generators. More precisely, we assert the following.

Proposition 6.6**.**

For any $n,m,p,q=0,1,2,\cdots$, we have

[TABLE]

Proof.

The first identity in (6.15) is obvious keeping in mind the convention that $[\frac{j}{k}]=+\infty$ when $k=0$. While the second one, i.e., when $q=0$ and $p\geq 1$, can be derived making use of $H_{n}^{(p)}(z|\gamma)=(z+p{\gamma}\partial_{z}^{p-1})^{n}\{1\}$ in [5, Eq (6), p. 18]. Indeed, we have $H_{n,m}^{(p,0)}(z,w|\gamma)=w^{m}H_{n}^{(p)}(z|\gamma)=(z+p{\gamma}\partial_{z}^{p-1})^{n}\{w^{m}\}.$ The last identity, corresponding to $p\geq 1$ and $q\geq 1$, can be handled by induction on $n$. Indeed, we have $H_{n,m}^{(p,q)}(z,w|\gamma)=(z+p\gamma\partial_{z}^{p-1}\partial_{w}^{q})H_{n-1,m}^{(p,q)}(z,w|\gamma),$ and therefore, $H_{n,m}^{(p,q)}(z,w|\gamma)=(z+p\gamma\partial_{z}^{p-1}\partial_{w}^{q})^{n}H_{0,m}^{(p,q)}(z,w|\gamma),$ where $H_{0,m}^{(p,q)}(z,w|\gamma)=w^{m}$. Then, we have $H_{n,m}^{(p,q)}(z,w|\gamma)=(z+p\gamma\partial_{z}^{p-1}\partial_{w}^{q})^{n}\{w^{m}\}.$ ∎

Remark 6.7**.**

The analogues of second and third recursion formulas, with respect to $z$ variable, read $H_{n,m}^{(p,q)}(z,w|\gamma)=(w+q\gamma\partial_{w}^{q-1})^{m}\{z^{n}\}$ and $H_{n,m}^{(p,q)}(z,w|\gamma)=(w+q\gamma\partial_{z}^{p}\partial_{w}^{q-1})^{m}\{z^{n}\}$, respectively, and follows by the use of symmetry identity.

Accordingly, we see that

[TABLE]

This can be reformulated as follows thanks to $H_{0,0}^{(p,q)}(z,w|\gamma)=1$.

Corollary 6.8**.**

We have

[TABLE]

The next assertion is a recursion relation with respect to parameters $p$ and $q$.

Proposition 6.9**.**

We have

[TABLE]

Proof.

Making appeal of the generating function $R^{p,q}_{\gamma}(z,w|u,v)$ in (3.3), we get

[TABLE]

Therefore, in view of (5.2), we obtain

[TABLE]

The result (6.17) readily follows by identification. ∎

The previous recursion relation can be shown to equivalent to the operational formula (6.18) below.

Proposition 6.10**.**

We have

[TABLE]

Proof.

Clearly (6.19) follows from (6.18) by symmetry, while (6.20) is immediate consequence of combining (6.18) and (6.19) keeping in mind that $(\partial_{z}-1)\partial_{z}^{p}\partial_{w}^{q}$ and $(\partial_{w}-1)\partial_{z}^{p}\partial_{w}^{q}$ are commuting. We need only to prove (6.18). Starting from the definition of $H_{n,m}^{(p+1,q)}(z,w|\gamma)$ and using (6.4), we obtain

[TABLE]

∎

Remark 6.11**.**

*These recursion relations are new even restricting to the Gould–Hopper polynomials. *

7. Nielsen identities

In this section, we prove some summation formulas of Nielsen type for the polynomials $H_{n,m}^{(p,q)}$ which can be used to derive others addition formulas.

Theorem 7.1**.**

We have

[TABLE]

Proof.

We need to prove only the first identity. Indeed, by (3.1) we have

[TABLE]

Applying this fact twice for given $z$ and $z^{\prime}$, we find

[TABLE]

The last equality follows using [22, Eq. (1), p. 100]. Now, the substitution of $n$ by $n-k$ and $n^{\prime}$ by $n^{\prime}-j$ lead to

[TABLE]

The result in (7.1) follows by identification. ∎

A generalization of Theorem 7.1 is the following one which readily follows from (7.1), applied to $H_{n+n^{\prime},m+m^{\prime}}^{(p,q)}(z,w|\gamma)$, and (7.2). Set

[TABLE]

Proposition 7.2**.**

We have

[TABLE]

Remark 7.3**.**

*Notice that for $m=q=0$ in Proposition 7.2, we get the formula for the Gould-Hopper polynomials [20]. *

As immediate consequence, we obtain the following addition formula with respect to the variables $z$ and $w$.

Corollary 7.4**.**

We have

[TABLE]

Proof.

This readily follows from (7.4) by specifying $n^{\prime}=m^{\prime}=0$, and replacing $z$ by $z-z^{\prime}$ and $w$ by $w-w^{\prime}$. ∎

Corollary 7.5**.**

We have

[TABLE]

Proof.

It suffices to replace $z$ and $z^{\prime}$ by $\frac{z}{2}$, and $w$ and $w^{\prime}$ by $\frac{w}{2}$ in (7.5), and next applying (3.5). ∎

8. Connection to Gould-Hopper polynomials

The main aim here is to express the polynomials $H_{n,m}^{(p,q)}$ in terms of the Gould-Hopper polynomials $H_{n}^{(p)}$ and vice-versa. We begin by expressing $H_{n}^{(p)}$ in function of $H_{n,m}^{(p,q)}$.

Proposition 8.1**.**

We have

[TABLE]

Proof.

The expression of $H_{n}^{(p)}$ in terms of $H_{n,m}^{(p,q)}$ as given through (8.1) is in fact equivalent to the following

[TABLE]

which readily follows by identification process. Indeed, by taking $v=u$ in the generating function (3.3) and substituting there $n$ by $n-k$, we obtain

[TABLE]

The last equality follows making use of the generating function (3.4) for Gould–Hopper polynomials. ∎

Remark 8.2**.**

The specification of $p=q=1$, $\gamma=-1$ and $m=0$ in (8.1) (or (8.2)) provides us with a new expression of the holomrphic Hermite polynomials $H_{n}(z)$ in therms of the polyanalytic Ito–Hermite polynomials,

[TABLE]

Conversely, we assert the following.

Proposition 8.3**.**

We have

[TABLE]

Proof.

Using the generating function (3.3) and the fact that $e^{zu+wv+{\gamma}u^{p}v^{q}}=e^{zu+\frac{{\gamma}}{2}u^{p}v^{q}}e^{wv+\frac{{\gamma}}{2}u^{p}v^{q}}$, as well as the identity (5.3), it follows

[TABLE]

Replacing $n+ip$ by $n$ and $m+iq$ by $m$, we get

[TABLE]

This completes the proof of (8.3). ∎

9. Concluding remarks

The extension of the classical results valid for Hermite, Ito-Hermite and Gould–Hopper polynomials for the $(p,q)$ Gould–Hopper polynomials has been discussed and presented in an unified way. It seems that the considered polynomials will be a fundamental tool in studying the high order partial differential equation in (1.4). In fact, the obtained recurrence formulas, in the previous section, show that the polynomials $H_{n,m}^{(p,q)}$ satisfy certain partial differential equations, generalizing, somehow, those obtained for the real and complex Hermite type polynomials considered in this paper. Indeed, from (6.3) we have

[TABLE]

This mean that the polynomials $H_{n,m}^{(p,q)}$ are solutions for the heat $(p,q)$-differential equation $\partial_{\gamma}u=\partial_{z}^{p}\partial_{w}^{q}u.$ Another partial differential equation satisfied by $H_{n,m}^{(p,q)}$ follows making use of (6.1) and the recursion relation (6.8). Namely, we have

[TABLE]

This means that the polynomials $H_{n,m}^{(p,q)}$ are eigenfunctions of the partial differential operators $\Delta^{p,q}_{\gamma}:=z\partial_{z}+\gamma\partial_{z}^{p}\partial_{w}^{q}$ and $\widetilde{\Delta^{p,q}_{\gamma}}:=w\partial_{w}+\gamma\partial_{z}^{p}\partial_{w}^{q}$. Therefore they are solutions of the following system

[TABLE]

Subsequently and since the operator $\partial_{z}$ and $(w+\gamma\partial_{z}^{p}\partial_{w}^{q-1})$ commute, the polynomials $H_{n,m}^{(p,q)}$ obey to the following high order partial differential equation

[TABLE]

The last equation means also that the polynomials $H_{n,m}^{(p,q)}$ are eigenfunction of the operator $(z+\gamma\partial_{z}^{p-1}\partial_{w}^{q})(w+\gamma\partial_{z}^{p}\partial_{w}^{q-1})\partial_{z}\partial_{w},$ corresponding to the eigenvalue $nm$.