# On σ-type zero of Sheffer polynomials

## Abstract

The main object of this paper is to investigate some properties of σ-type polynomials in one and two variables.

MSC:33C65, 33E99, 44A45, 46G25.

## 1 Introduction

In 1945, Thorne [1] obtained an interesting characterization of Appell polynomials by means of the Stieltjes integral. Srivastava and Manocha [2] discussed the Appell sets and polynomials. Dattoli et al. [3] studied the properties of the Sheffer polynomials. Recently Pintér and Srivastava [4] gave addition theorems for the Appell polynomials and the associated classes of polynomial expansions and some cases have also been discussed by Srivastava and Choi [5] in their book.

Appell sets may be defined by the following equivalent condition: $\left\{{P}_{n}\left(x\right)\right\}$, $n=0,1,2,\dots$ is an Appell set [68] (${P}_{n}$ being of degree exactly n) if either

1. (i)

${P}_{n}^{\prime }\left(x\right)={P}_{n-1}\left(x\right)$, $n=0,1,2,\dots$ , or

2. (ii)

there exists a formal power series $A\left(t\right)={\sum }_{n=0}^{\mathrm{\infty }}{a}_{n}{t}^{n}$ (${a}_{0}\ne 0$) such that

$A\left(t\right)exp\left(xt\right)=\sum _{n=0}^{\mathrm{\infty }}{P}_{n}\left(x\right){t}^{n}.$

### Sheffer’s A-type classification

Let ${\varphi }_{n}\left(x\right)$ be a simple set of polynomials and let ${\varphi }_{n}\left(x\right)$ belong to the operator

$J\left(x,D\right)=\sum _{k=0}^{\mathrm{\infty }}{T}_{k}\left(x\right){D}^{k+1},$

with ${T}_{k}\left(x\right)$ of degree ≤k. If the maximum degree of the coefficients ${T}_{k}\left(x\right)$ is m, then the set ${\varphi }_{n}\left(x\right)$ is of Sheffer A-type m. If the degree of ${T}_{k}\left(x\right)$ is unbounded as $k\to \mathrm{\infty }$, we say that ${\varphi }_{n}\left(x\right)$ is of Sheffer A-type ∞.

### Polynomials of Sheffer A-type zero

Let ${\varphi }_{n}\left(x\right)$ be of Sheffer A-type zero. Then ${\varphi }_{n}\left(x\right)$ belong to the operator

$J\left(D\right)=\sum _{k=0}^{\mathrm{\infty }}{c}_{k}{D}^{k+1},$

in which ${c}_{k}$ are constants. Here ${c}_{0}\ne 0$ and $J{\varphi }_{n}={\varphi }_{n-1}$. Furthermore, since ${c}_{k}$ are independent of x for every k, a function $J\left(t\right)$ exists with the formal power series expansion

$J\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}{c}_{k}{t}^{k+1},\phantom{\rule{1em}{0ex}}{c}_{0}\ne 0.$

Let $H\left(t\right)$ be the formal inverse of $J\left(t\right)$; that is,

$H\left(J\left(t\right)\right)=J\left(H\left(t\right)\right)=t.$

Theorem (Rainville [9])

A necessary and sufficient condition that ${\varphi }_{n}\left(x\right)$ be of Sheffer A-type zero is that ${\varphi }_{n}\left(x\right)$ possess the generating function indicated in

$A\left(t\right)exp\left(xH\left(t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n}\left(x\right){t}^{n},$

in which $H\left(t\right)$ and $A\left(t\right)$ have (formal) expansions

$H\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{h}_{n}{t}^{n+1},\phantom{\rule{1em}{0ex}}{h}_{0}\ne 0,\phantom{\rule{2em}{0ex}}A\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}{t}^{n},\phantom{\rule{1em}{0ex}}{a}_{0}\ne 0.$

Theorem (Al-Salam and Verma [10])

Let $\left\{{P}_{n}\left(x\right)\right\}$ be a polynomial set. In order for $\left\{{P}_{n}\left(x\right)\right\}$ to be a Sheffer A-type zero, it is necessary and sufficient that there exist (formal) power series

$H\left(t\right)=\sum _{j=1}^{\mathrm{\infty }}{h}_{j}{t}^{j},\phantom{\rule{1em}{0ex}}{h}_{1}\ne 0,\phantom{\rule{2em}{0ex}}{A}_{s}\left(t\right)=\sum _{j=0}^{\mathrm{\infty }}{a}_{j}^{\left(s\right)}{t}^{j}\phantom{\rule{1em}{0ex}}\left(\mathit{\text{not all}}{a}_{0}^{\left(s\right)}\mathit{\text{are zero}}\right)$

and

$\sum _{j=1}^{r}{A}_{j}\left(t\right)exp\left(xH\left({\epsilon }_{j}t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{P}_{n}\left(x\right){t}^{n},$

where

$J\left(D\right){P}_{n}\left(x\right)={P}_{n-r}\left(x\right)\phantom{\rule{1em}{0ex}}\left(n=r,r+1,\dots \right)\mathit{\text{where}}J\left(D\right)=\sum _{k=0}^{\mathrm{\infty }}{a}_{k}{D}^{k+r},{a}_{0}\ne 0$

and r is a fixed positive integer. The function $A\left(t\right)$ may be called the determining function for the set $\left\{{P}_{n}\left(x\right)\right\}$.

### Polynomial of σ-type zero [9, 11]

Let $\left\{{p}_{n}\left(x\right)\right\}$ be a simple set of polynomials that belongs to the operator

$\begin{array}{c}J\left(x,\sigma \right)=\sum _{k=0}^{\mathrm{\infty }}{T}_{k}\left(x\right){\sigma }^{k+1},\hfill \\ \sigma =D\prod _{i=1}^{q}\left(xD+{b}_{i}-1\right),\phantom{\rule{2em}{0ex}}D=\frac{d}{dx},\phantom{\rule{2em}{0ex}}\left(J\left(x,\sigma \right){p}_{n}\left(x\right)={p}_{n-1}\left(x\right)\right),\hfill \end{array}$

where ${b}_{i}$ are constants, not equal to zero or a negative integer, and ${T}_{k}\left(x\right)$ are polynomials of degree ≤k. We can say that this set is of σ-type m if the maximum degree of ${T}_{k}\left(x\right)$ is m, $m=0,1,2,\dots$ .

A necessary and sufficient condition that ${\varphi }_{n}\left(x\right)$ be of σ-type zero, with

$\sigma =D\prod _{i=1}^{q}\left(xD+{b}_{i}-1\right),$

is that ${\varphi }_{n}\left(x\right)$ possess the generating function

$A{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left(t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{n}\left(x\right){t}^{n},$

in which $H\left(t\right)$ and $A\left(t\right)$ have (formal) expansions

$H\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{h}_{n}{t}^{n+1},\phantom{\rule{1em}{0ex}}{h}_{0}\ne 0,$

and

$A\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}{t}^{n},\phantom{\rule{1em}{0ex}}{a}_{0}\ne 0.$

Since ${\varphi }_{n}\left(x\right)$ belongs to the operator $J\left(\sigma \right)={\sum }_{k=0}^{\mathrm{\infty }}{c}_{k}{\sigma }^{k+1}$, where ${c}_{k}$ are constant and ${c}_{0}\ne 0$.

## 2 Main results

Theorem 1 If ${p}_{n}\left(x\right)$ is a polynomial set, then ${p}_{n}\left(x\right)$ is of σ-type zero with $\sigma =D{\prod }_{m=1}^{q}\left(xD+{b}_{m}-1\right)$. It is necessary and sufficient condition that there exist formal power series

$H\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{h}_{n}{t}^{n+1},\phantom{\rule{1em}{0ex}}{h}_{0}\ne 0,$

and

${A}_{i}\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}^{\left(i\right)}{t}^{n}\phantom{\rule{1em}{0ex}}\left(\mathit{\text{not all}}{a}_{0}^{\left(i\right)}\mathit{\text{are zero}}\right)$

such that

$\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n},$
(1)

where $\theta =xD$.

Proof Let ${y}_{i}{=}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};{z}_{i}\right)$, where $i=1,2,\dots ,r$, be a solution of the following differential equation:

$\left[\theta \prod _{m=1}^{q}\left(xD+{b}_{m}-1\right)-{z}_{i}\right]{y}_{i}=0,\phantom{\rule{2em}{0ex}}\theta =xD,\phantom{\rule{2em}{0ex}}D=\frac{d}{dx}.$

On substituting ${z}_{i}=xH\left({\epsilon }_{i}t\right)$ and keeping t as a constant, where

$\sigma =D\prod _{m=1}^{q}\left(xD+{b}_{m}-1\right),\phantom{\rule{2em}{0ex}}\theta =xD,$

we get

$\left[xD\prod _{m=1}^{q}\left(\theta +{b}_{m}-1\right)-xH\left({z}_{i}\right)\right]{y}_{i}=0.$

This can also be written as

$\sigma {y}_{i}=H\left({\epsilon }_{i}t\right){y}_{i}$

or

${\sigma }_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)=H{\left({\epsilon }_{i}t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right).$

Operating $J\left(\sigma \right)$ on both sides of Equation (1) yields

$\begin{array}{rl}J\left(\sigma \right)\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n}& =J\left(\sigma \right)\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)\\ =\sum _{i=1}^{r}{A}_{i}\left(t\right)J{\left(H\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)\\ =t\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n}\\ =\sum _{n=1}^{\mathrm{\infty }}{p}_{n-1}\left(x\right){t}^{n}.\end{array}$

Therefore, $J\left(\sigma \right){p}_{0}\left(x\right)=0$ and $J\left(\sigma \right){p}_{n}\left(x\right)={p}_{n-1}\left(x\right)$, $n\ge 1$.

Since $J\left(\sigma \right)$ is independent of x, using the definition of σ-type [9, 11], we arrive at the conclusion that ${p}_{n}\left(x\right)$ is σ-type zero.

Conversely, suppose ${p}_{n}\left(x\right)$ is of σ-type zero and belongs to the operator $J\left(\sigma \right)$. Now ${q}_{n}\left(x\right)$ is a simple set of polynomials, we can write

${\sum _{i=1}^{r}}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n},$
(2)

where ${\epsilon }_{1},{\epsilon }_{2},\dots ,{\epsilon }_{r}$ are the roots of unity.

Since ${q}_{n}\left(x\right)$ is a simple set, there exists a sequence ${c}_{k}$ [10], independent of n, such that

${p}_{n}\left(x\right)=\sum _{k=0}^{n}{c}_{n-k}{q}_{k}\left(x\right)$

and

$\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n}=\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{n}{c}_{n-k}{q}_{k}\left(x\right){t}^{n}.$

On replacing n by $n+k$, this becomes

$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n}& =\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}{c}_{n}{q}_{k}\left(x\right){t}^{n+k}\\ =\sum _{k=0}^{\mathrm{\infty }}{q}_{k}\left(x\right){t}^{k}\sum _{n=0}^{\mathrm{\infty }}{c}_{n}{t}^{n}.\end{array}$

Setting ${c}_{n}={a}_{n}^{\left(i\right)}$ (i is independent of n, where $i=1,2,\dots ,r$), this becomes

$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n}& =\sum _{k=0}^{\mathrm{\infty }}{q}_{k}\left(x\right){t}^{k}\sum _{n=0}^{\mathrm{\infty }}{a}_{n}^{\left(i\right)}{t}^{n},\phantom{\rule{1em}{0ex}}\text{by using Equation (2), we get}\\ =\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right).\end{array}$

This completes the proof. □

Theorem 2 A necessary and sufficient condition that ${p}_{n}\left(x\right)$ be of σ-type zero and there exist a sequence ${h}_{k}$, independent of x and n, such that

$\sum _{i=1}^{r}{\epsilon }_{i}^{n}{h}_{n-1}\psi \left({\epsilon }_{i}t\right)=\sigma {p}_{n}\left(x\right),$
(3)

where $\psi \left({\epsilon }_{i}t\right)={A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)$.

Proof If ${p}_{n}\left(x\right)$ is of σ-type zero, then it follows from Theorem 1 that

$\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x\right){t}^{n}=\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right).$

This can be written as

$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}\sigma {p}_{n}\left(x\right){t}^{n}& =\sum _{i=1}^{r}{A}_{i}\left(t\right){\sigma }_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)\\ =\sum _{i=1}^{r}H\left({\epsilon }_{i}t\right){A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)\\ =\sum _{i=1}^{r}\left(\sum _{n=0}^{\mathrm{\infty }}{h}_{n}{\epsilon }_{i}^{n+1}{t}^{n+1}\right){A}_{i}{\left(t\right)}_{0}{F}_{q}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{q};xH\left({\epsilon }_{i}t\right)\right)\\ =\sum _{n=1}^{\mathrm{\infty }}\sum _{i=1}^{r}\left({\epsilon }_{i}^{n}{h}_{n-1}\right)\psi \left({\epsilon }_{i}t\right){t}^{n}.\end{array}$

Thus

$\sigma {p}_{n}\left(x\right)=\sum _{i=1}^{r}{\epsilon }_{i}^{n}{h}_{n-1}\psi \left({\epsilon }_{i}t\right).$

This completes the proof. □

## 3 Sheffer polynomials in two variables [12]

Let ${p}_{n}\left(x,y\right)$ be of σ-type zero. Then ${p}_{n}\left(x,y\right)$ belongs to an operator $J\left(\sigma \right)={\sum }_{k=0}^{\mathrm{\infty }}{c}_{k}{\sigma }^{k+1}$, in which ${c}_{k}$ are constants and ${c}_{0}\ne 0$.

Since

$J\left(\sigma \right){p}_{n}\left(x,y\right)={p}_{n-1}\left(x,y\right),\phantom{\rule{1em}{0ex}}n\ge 1,$

where

$\begin{array}{c}{D}_{x}=\frac{\partial }{\partial x},\phantom{\rule{2em}{0ex}}{D}_{y}=\frac{\partial }{\partial y},\phantom{\rule{2em}{0ex}}\theta =x\frac{\partial }{\partial x},\phantom{\rule{2em}{0ex}}\varphi =y\frac{\partial }{\partial y},\hfill \\ {\sigma }_{x}={D}_{x}\prod _{m=1}^{p}\left(\theta +{b}_{m}-1\right),\phantom{\rule{2em}{0ex}}{\sigma }_{y}={D}_{y}\prod _{s=1}^{q}\left(\theta +{b}_{s}-1\right),\hfill \end{array}$

and

$J\left(\left(G+H\right)\left(t\right)\right)=\left(\left(G+H\right)J\left(t\right)\right)=t,\phantom{\rule{2em}{0ex}}\sigma ={\sigma }_{x}+{\sigma }_{y}.$

Theorem 3 A necessary and sufficient condition that ${p}_{n}\left(x,y\right)$ be of σ-type zero, with

${\sigma }_{x}={D}_{x}\prod _{m=1}^{p}\left(\theta +{b}_{m}-1\right),\phantom{\rule{2em}{0ex}}{\sigma }_{y}={D}_{y}\prod _{s=1}^{q}\left(\theta +{b}_{s}-1\right),\phantom{\rule{2em}{0ex}}\sigma ={\sigma }_{x}+{\sigma }_{y},$

is that ${p}_{n}\left(x,y\right)$ possess a generating function in

$\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x,y\right){t}^{n},$
(4)

in which

$\begin{array}{c}G\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{g}_{n}{t}^{n+1},\phantom{\rule{1em}{0ex}}{g}_{0}\ne 0,\hfill \\ H\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{h}_{n}{t}^{n+1},\phantom{\rule{1em}{0ex}}{h}_{0}\ne 0,\hfill \\ {A}_{i}\left(t\right)=\sum _{n=0}^{\mathrm{\infty }}{a}_{n}^{\left(i\right)}{t}^{n}\phantom{\rule{1em}{0ex}}\left(\mathit{\text{not all}}{a}_{0}^{\left(i\right)}\mathit{\text{are zero}}\right)\hfill \end{array}$

and i is independent of n.

Proof Let ${u}_{i}{=}_{0}{F}_{p}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};{z}_{i}\right)$ and ${v}_{i}{=}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};{w}_{i}\right)$ be the solutions of the following differential equations:

$\left[{\theta }_{z}\prod _{m=1}^{p}\left({\theta }_{z}+{b}_{m}-1\right)-{z}_{i}\right]{u}_{i}=0,\phantom{\rule{2em}{0ex}}{\theta }_{z}=z\frac{\partial }{\partial z},$

and

$\left[{\varphi }_{w}\prod _{s=1}^{q}\left({\varphi }_{w}+{c}_{s}-1\right)-{z}_{i}\right]{w}_{i}=0,\phantom{\rule{2em}{0ex}}{\varphi }_{w}=w\frac{\partial }{\partial w}.$

On substituting ${z}_{i}=xG\left({\epsilon }_{i}t\right)$, ${w}_{i}=yH\left({\epsilon }_{i}t\right)$ and keeping t as a constant, where $\theta =x\frac{\partial }{\partial x}={\theta }_{z}$, $\varphi =y\frac{\partial }{\partial y}={\varphi }_{w}$, we get

$\theta \prod _{m=1}^{p}\left(\theta +{b}_{m}-1\right){u}_{i}=xG\left({\epsilon }_{i}t\right){u}_{i}$

and

$\varphi \prod _{s=1}^{q}\left(\varphi +{c}_{s}-1\right){w}_{i}=yH\left({\epsilon }_{i}t\right){w}_{i}.$

This can also be written as

$\begin{array}{c}{\sigma }_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)\hfill \\ \phantom{\rule{1em}{0ex}}={\left\{G\left({\epsilon }_{i}t\right)+H\left({\epsilon }_{i}t\right)\right\}}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right).\hfill \end{array}$

Operating $J\left(\sigma \right)$ on both sides of Equation (4) yields

$\begin{array}{r}J\left(\sigma \right)\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x,y\right){t}^{n}\\ \phantom{\rule{1em}{0ex}}=J\left(\sigma \right)\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)\\ \phantom{\rule{1em}{0ex}}=\sum _{i=1}^{r}{A}_{i}\left(t\right)J{\left(\left(G+H\right)\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{p}{\left[-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right]}_{0}{F}_{q}\left[-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right]\\ \phantom{\rule{1em}{0ex}}=t\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x,y\right){t}^{n}\\ \phantom{\rule{1em}{0ex}}=\sum _{n=1}^{\mathrm{\infty }}{p}_{n-1}\left(x,y\right){t}^{n}.\end{array}$

Therefore, $J\left(\sigma \right){p}_{0}\left(x,y\right)=0$ and $J\left(\sigma \right){p}_{n}\left(x,y\right)={p}_{n-1}\left(x,y\right)$, $n\ge 1$.

Since $J\left(\sigma \right)$ is independent of x and y, thus we arrive at the conclusion that ${p}_{n}\left(x,y\right)$ is of σ-type zero.

Conversely, suppose ${p}_{n}\left(x,y\right)$ is of σ-type zero and belongs to the operator $J\left(\sigma \right)$. Now ${q}_{n}\left(x,y\right)$ is a simple set of polynomials. We can write

${\sum _{i=1}^{r}}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)=\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x,y\right){t}^{n}.$
(5)

Since ${q}_{n}\left(x,y\right)$ is a simple set, there exists a sequence ${c}_{k}$, independent of n, such that

${p}_{n}\left(x,y\right)=\sum _{k=0}^{n}{c}_{n-k}{q}_{k}\left(x,y\right)$

and

$\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x,y\right){t}^{n}=\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{n}{c}_{n-k}{q}_{k}\left(x,y\right){t}^{n}.$

On replacing n by $n+k$, this becomes

$\begin{array}{c}=\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}{c}_{n}{q}_{k}\left(x,y\right){t}^{n+k}\hfill \\ =\sum _{k=0}^{\mathrm{\infty }}{q}_{k}\left(x,y\right){t}^{k}\sum _{n=0}^{\mathrm{\infty }}{c}_{n}{t}^{n}.\hfill \end{array}$

Setting ${c}_{n}={a}_{n}^{\left(i\right)}$ (i is independent of n, where $i=1,2,\dots ,r$), this becomes

$\begin{array}{c}=\sum _{k=0}^{\mathrm{\infty }}{q}_{k}\left(x,y\right){t}^{k}\sum _{n=0}^{\mathrm{\infty }}{a}_{n}^{\left(i\right)}{t}^{n}\hfill \\ =\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right).\hfill \end{array}$

This completes the proof. □

Theorem 4 A necessary and sufficient condition that ${p}_{n}\left(x,y\right)$ be of σ-type zero and there exist sequences ${g}_{k}$ and ${h}_{k}$ , independent of x, y and n, such that

$\sum _{i=1}^{r}{\epsilon }_{i}^{n}\left({g}_{n-1}+{h}_{n-1}\right)\upsilon \left({\epsilon }_{i}t\right)=\sigma {p}_{n}\left(x,y\right),$
(6)

where $\upsilon \left({\epsilon }_{i}t\right)={A}_{i}{\left(t\right)}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)$.

Proof If ${p}_{n}\left(x,y\right)$ is of σ-type zero, then it follows from Theorem 3 that

$\sum _{n=0}^{\mathrm{\infty }}{p}_{n}\left(x,y\right){t}^{n}=\sum _{i=1}^{r}{A}_{i}{\left(t\right)}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right).$

This can be written as

$\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}\sigma {p}_{n}\left(x,y\right){t}^{n}=& \sum _{i=1}^{r}{A}_{i}\left(t\right){\sigma }_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)\\ =& \sum _{i=1}^{r}\left(G+H\right)\left({\epsilon }_{i}t\right){A}_{i}{\left(t\right)}_{0}{F}_{p}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)\\ {×}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)\\ =& \sum _{i=1}^{r}\left(\sum _{n=0}^{\mathrm{\infty }}\left({g}_{n}+{h}_{n}\right){\epsilon }_{i}^{n+1}{t}^{n+1}\right){A}_{i}{\left(t\right)}_{0}{F}_{p}\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)\\ {×}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)\\ =& \sum _{n=1}^{\mathrm{\infty }}\sum _{i=1}^{r}\left({\epsilon }_{i}^{n}\left({g}_{n-1}+{h}_{n-1}\right)\right)\upsilon \left({\epsilon }_{i}t\right){t}^{n}.\end{array}$

Thus

$\sigma {p}_{n}\left(x,y\right)=\sum _{i=1}^{r}{\epsilon }_{i}^{n}\left({g}_{n-1}+{h}_{n-1}\right)\upsilon \left({\epsilon }_{i}t\right),$

where $\upsilon \left({\epsilon }_{i}t\right)={A}_{i}{\left(t\right)}_{0}{F}_{p}{\left(-;{b}_{1},{b}_{2},\dots ,{b}_{p};xG\left({\epsilon }_{i}t\right)\right)}_{0}{F}_{q}\left(-;{c}_{1},{c}_{2},\dots ,{c}_{q};yH\left({\epsilon }_{i}t\right)\right)$. This completes the proof. □

## References

1. Thorne CJ: A property of Appell sets. Am. Math. Mon. 1945, 52: 191–193. 10.2307/2305676

2. Srivastava HM, Manocha HL: A Treatise on Generating Functions. Wiley, New York; 1984.

3. Dattoli G, Migliorati M, Srivastava HM: Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials. Math. Comput. Model. 2007, 45(9–10):1033–1041. 10.1016/j.mcm.2006.08.010

4. Pintér Á, Srivastava HM: Addition theorems for the Appell polynomials and the associated classes of polynomial expansions. Aequ. Math. 2012. doi:10.1007/s00010–012–0148–8

5. Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.

6. Galiffa, JD: The Sheffer B-type 1 orthogonal polynomial sequences. Ph.D. Dissertation, University of Central Florida (2009)

7. Goldberg JL:On the Sheffer A-type of polynomials generated by $A\left(t\right)\psi \left(xB\left(t\right)\right)$. Proc. Am. Math. Soc. 1966, 17: 170–173.

8. Sheffer IM: Note on Appell polynomials. Bull. Am. Math. Soc. 1945, 51: 739–744. 10.1090/S0002-9904-1945-08437-7

9. Rainville ED: Special Functions. Macmillan Co., New York; 1960.

10. Al Salam WA, Verma A: Generalized Sheffer polynomials. Duke Math. J. 1970, 37: 361–365. 10.1215/S0012-7094-70-03746-4

11. Mc Bride EB: Obtaining Generating Functions. Springer, New York; 1971.

12. Alidad, B: On some problems of special functions and structural matrix analysis. Ph.D. Dissertation, Aligarh Muslim University (2008)

## Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are indebted to the referees for their valuable suggestions which led to a better presentation of paper.

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Correspondence to Ajay K Shukla.

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The authors contributed equally and significantly in writing this article. The authors read and approved the final manuscript.

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Shukla, A.K., Rapeli, S.J. & Shah, P.V. On σ-type zero of Sheffer polynomials. J Inequal Appl 2013, 241 (2013). https://doi.org/10.1186/1029-242X-2013-241