Theory and Modern Applications

# Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials

## Abstract

In this paper, we introduce a unified family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials. We obtain some symmetry identities between these polynomials and the generalized sum of integer powers. We give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.

MSC:11B68, 33C05.

## 1 Introduction

Recently, Khan et al. [1] introduced the Hermite-based Appell polynomials via the generating function

$\mathcal{G}\left(x,y,z;t\right)=A\left(t\right)exp\left(\mathcal{M}t\right),$

where

$\mathcal{M}=x+2y\frac{\mathrm{âˆ‚}}{\mathrm{âˆ‚}x}+3z\frac{{\mathrm{âˆ‚}}^{2}}{\mathrm{âˆ‚}{x}^{2}}$

is the multiplicative operator of the 3-variable Hermite polynomials, which are defined by

$exp\left(xt+y{t}^{2}+z{t}^{3}\right)=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{H}_{n}^{\left(3\right)}\left(x,y,z\right)\frac{{t}^{n}}{n!}$
(1.1)

and

By using the Berry decoupling identity,

${e}^{A+B}={e}^{{m}^{2}/12}{e}^{\left(\left(\frac{âˆ’m}{2}\right){A}^{1/2}+A\right)}{e}^{B},\phantom{\rule{1em}{0ex}}\left[A,B\right]=m{A}^{1/2}$

they obtained the generating function of the Hermite-based Appell polynomials ${}_{H}A_{n}\left(x,y,z\right)$ as

$\mathcal{G}\left(x,y,z;t\right)=A\left(t\right)exp\left(xt+y{t}^{2}+z{t}^{3}\right)={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{A}_{n}\left(x,y,z\right)\frac{{t}^{n}}{n!}.$

Letting $A\left(t\right)=\frac{t}{{e}^{t}âˆ’1}$, they defined Hermite-Bernoulli polynomials ${}_{H}B_{n}\left(x,y,z\right)$ by

$\frac{t}{{e}^{t}âˆ’1}exp\left(xt+y{t}^{2}+z{t}^{3}\right)={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{B}_{n}\left(x,y,z\right)\frac{{t}^{n}}{n!},\phantom{\rule{1em}{0ex}}|t|<2\mathrm{Ï€}.$

For $A\left(t\right)=\frac{2}{{e}^{t}+1}$, they defined Hermite-Euler polynomials ${}_{H}E_{n}\left(x,y,z\right)$ by

$\frac{2}{{e}^{t}+1}exp\left(xt+y{t}^{2}+z{t}^{3}\right)={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{E}_{n}\left(x,y,z\right)\frac{{t}^{n}}{n!},\phantom{\rule{1em}{0ex}}|t|<\mathrm{Ï€}$

and for $A\left(t\right)=\frac{2t}{{e}^{t}+1}$, they defined Hermite-Genocchi polynomials ${}_{H}G_{n}\left(x,y,z\right)$ by

$\frac{2t}{{e}^{t}+1}exp\left(xt+y{t}^{2}+z{t}^{3}\right)={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{G}_{n}\left(x,y,z\right)\frac{{t}^{n}}{n!},\phantom{\rule{1em}{0ex}}|t|<\mathrm{Ï€}.$

Recently, the author considered the following unification of the Apostol-Bernoulli, Euler and Genocchi polynomials

$\begin{array}{r}{f}_{a,b}^{\left(\mathrm{Î±}\right)}\left(x;t;k,\mathrm{Î²}\right):={\left(\frac{{2}^{1âˆ’k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{t}âˆ’{a}^{b}}\right)}^{\mathrm{Î±}}{e}^{xt}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x;k,a,b\right)\frac{{t}^{n}}{n!}\\ \phantom{\rule{1em}{0ex}}\left(kâˆˆ{\mathbb{N}}_{0};a,bâˆˆ\mathbb{R}\mathrm{âˆ–}\left\{0\right\};\mathrm{Î±},\mathrm{Î²}âˆˆ\mathbb{C}\right)\end{array}$

and obtained the explicit representation of this unified family, in terms of Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given in [2]. Note that the family of polynomials ${P}_{n,\mathrm{Î²}}^{\left(1\right)}\left(x,y,z;k,a,b\right)$ was investigated in [3].

We organize the paper as follows.

In Section 2, we introduce the unification of the Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials ${}_{H}P_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)$ and give summation formulas for this unification. In Section 3, we obtain some symmetry identities for these polynomials. In Section 4, we give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.

## 2 Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials

In this paper, we consider the following general class of polynomials:

$\begin{array}{r}{f}_{a,b}^{\left(\mathrm{Î±}\right)}\left(x,y,z;t;k,\mathrm{Î²}\right):={\left(\frac{{2}^{1âˆ’k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{t}âˆ’{a}^{b}}\right)}^{\mathrm{Î±}}{e}^{xt+y{t}^{2}+z{t}^{3}}={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\\ \phantom{\rule{1em}{0ex}}\left(kâˆˆ{\mathbb{N}}_{0};a,bâˆˆ\mathbb{R}\mathrm{âˆ–}\left\{0\right\};\mathrm{Î±},\mathrm{Î²}âˆˆ\mathbb{C}\right).\end{array}$
(2.1)

For the existence of the expansion, we need

1. (i)

$|t|<2\mathrm{Ï€}$ when $\mathrm{Î±}âˆˆ\mathbb{C}$, $k=1$ and ${\left(\frac{\mathrm{Î²}}{a}\right)}^{b}=1$; $|t|<2\mathrm{Ï€}$ when $\mathrm{Î±}âˆˆ{\mathbb{N}}_{0}$, $k=2,3,â€¦$ and ${\left(\frac{\mathrm{Î²}}{a}\right)}^{b}=1$; $|t|<|blog\left(\frac{\mathrm{Î²}}{a}\right)|$ when $\mathrm{Î±}âˆˆ{\mathbb{N}}_{0}$, $kâˆˆ\mathbb{N}$ and (or ); $x,y,zâˆˆ\mathbb{R}$, $\mathrm{Î²}âˆˆ\mathbb{C}$, $a,bâˆˆ\mathbb{C}/\left\{0\right\}$; ${1}^{\mathrm{Î±}}:=1$;

2. (ii)

$|t|<\mathrm{Ï€}$ when ${\left(\frac{\mathrm{Î²}}{a}\right)}^{b}=âˆ’1$; $|t|<|blog\left(\frac{\mathrm{Î²}}{a}\right)|$ when ; $x,y,zâˆˆ\mathbb{R}$, $k=0$, $\mathrm{Î±},\mathrm{Î²}âˆˆ\mathbb{C}$, $a,bâˆˆ\mathbb{C}/\left\{0\right\}$; ${1}^{\mathrm{Î±}}:=1$;

3. (iii)

$|t|<\mathrm{Ï€}$ when $\mathrm{Î±}âˆˆ{\mathbb{N}}_{0}$ and ${\left(\frac{\mathrm{Î²}}{a}\right)}^{b}=âˆ’1$; $x,y,zâˆˆ\mathbb{R}$, $kâˆˆ\mathbb{N}$, $\mathrm{Î²}âˆˆ\mathbb{C}$, $a,bâˆˆ\mathbb{C}/\left\{0\right\}$; ${1}^{\mathrm{Î±}}:=1$,

where $w=|w|{e}^{i\mathrm{Î¸}}$, $âˆ’\mathrm{Ï€}â‰¤\mathrm{Î¸}<\mathrm{Ï€}$ and $log\left(w\right)=log\left(|w|\right)+i\mathrm{Î¸}$.

For $k=a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$ in (2.1), we define the following.

Definition 2.1 Let $\mathrm{Î±}âˆˆ{\mathbb{N}}_{0}$, Î» be an arbitrary (real or complex) parameter and $x,y,zâˆˆ\mathbb{R}$. The Hermite-based generalized Apostol-Bernoulli polynomials are defined by

It is clear that

${}_{H}P_{n,\mathrm{Î»}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;1,1,1\right){=}_{H}{\mathcal{B}}_{n}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right).$

Some special cases of the Hermite-based generalized Apostol-Bernoulli polynomials (some of which are definition) are listed below:

• ${}_{H}\mathcal{B}_{n}^{\left(1\right)}\left(x,y,z;\mathrm{Î»}\right):{=}_{H}{\mathcal{B}}_{n}\left(x,y,z;\mathrm{Î»}\right)$ is called Hermite-based Apostol-Bernoulli polynomials.

• ${}_{H}\mathcal{B}_{n}\left(x,y,z;1\right){=}_{H}{B}_{n}\left(x,y,z\right)$ is the Hermite-Bernoulli polynomials.

• ${}_{H}\mathcal{B}_{n}\left(x,0,0;\mathrm{Î»}\right):={\mathcal{B}}_{n}\left(x;\mathrm{Î»}\right)$ is the Apostol-Bernoulli polynomials (see [4â€“7]). When $\mathrm{Î»}=1$, we have the classical Bernoulli polynomials.

• ${\mathcal{B}}_{n}\left(0;\mathrm{Î»}\right):={\mathcal{B}}_{n}\left(\mathrm{Î»}\right)$ are the Apostol-Bernoulli numbers. $\mathrm{Î»}=1$ gives the classical Bernoulli numbers.

Setting $k+1=âˆ’a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$ in (2.1), we get the following.

Definition 2.2 Let Î± and Î» () be an arbitrary (real or complex) parameter and $x,y,zâˆˆ\mathbb{R}$. The Hermite-based generalized Apostol-Euler polynomials are defined by

Obviously, we have

${}_{H}P_{n,\mathrm{Î»}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;0,âˆ’1,1\right){=}_{H}{\mathcal{E}}_{n}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right).$

Some special cases of the Hermite-based generalized Apostol-Euler polynomials (some of which are definition) are listed below:

• ${}_{H}\mathcal{E}_{n}^{\left(1\right)}\left(x,y,z;\mathrm{Î»}\right):{=}_{H}{\mathcal{E}}_{n}\left(x,y,z;\mathrm{Î»}\right)$ is called Hermite-based Apostol-Euler polynomials.

• ${}_{H}\mathcal{E}_{n}\left(x,y,z;1\right){=}_{H}{E}_{n}\left(x,y,z\right)$ is the Hermite-Euler polynomials.

• ${}_{H}\mathcal{E}_{n}\left(x,0,0;\mathrm{Î»}\right):={\mathcal{E}}_{n}\left(x;\mathrm{Î»}\right)$ is the Apostol-Euler polynomials (see [8]). For $\mathrm{Î»}=1$, we have the classical Euler polynomials.

• ${2}^{n}{\mathcal{E}}_{n}\left(\frac{1}{2};\mathrm{Î»}\right):={\mathcal{E}}_{n}\left(\mathrm{Î»}\right)$ are the Apostol-Euler numbers. The case $\mathrm{Î»}=1$ gives the classical Euler numbers.

Choosing $k=âˆ’2a=b=1$ and $2\mathrm{Î²}=\mathrm{Î»}$ in (2.1), we define the following.

Definition 2.3 Let Î± and Î» () be an arbitrary (real or complex) parameter and $x,y,zâˆˆ\mathbb{R}$. The Hermite-based generalized Apostol-Genocchi polynomials are defined by

It is easily seen that

${}_{H}P_{n,\frac{\mathrm{Î»}}{2}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;1,\frac{âˆ’1}{2},1\right){=}_{H}{\mathcal{G}}_{n}^{\mathrm{Î±}}\left(x,y,z;\mathrm{Î»}\right).$

Some special cases of the Hermite-based generalized Apostol-Genocchi polynomials (some of which are definition) are listed below:

• ${}_{H}\mathcal{G}_{n}^{\left(1\right)}\left(x,y,z;\mathrm{Î»}\right):{=}_{H}{\mathcal{G}}_{n}\left(x,y,z;\mathrm{Î»}\right)$ is called Hermite-based Apostol-Genocchi polynomials.

• ${}_{H}\mathcal{G}_{n}\left(x,y,z;1\right){=}_{H}{G}_{n}\left(x,y,z\right)$ is the Hermite-Genocchi polynomials.

• ${}_{H}\mathcal{G}_{n}\left(x,0,0;\mathrm{Î»}\right):={\mathcal{G}}_{n}\left(x;\mathrm{Î»}\right)$ is the Apostol-Genocchi polynomials (see [9, 10]). When $\mathrm{Î»}=1$, we have the classical Genocchi polynomials.

• ${\mathcal{G}}_{n}\left(0;\mathrm{Î»}\right):={\mathcal{G}}_{n}\left(\mathrm{Î»}\right)$ are the Apostol-Genocchi numbers. $\mathrm{Î»}=1$ gives the classical Genocchi numbers.

Finally we define the unified Hermite-based Apostol polynomials by

$\begin{array}{r}{f}_{a,b}^{\left(1\right)}\left(x;t;k,\mathrm{Î²}\right):=\frac{{2}^{1âˆ’k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{t}âˆ’{a}^{b}}{e}^{xt+y{t}^{2}+z{t}^{3}}={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\\ \phantom{\rule{1em}{0ex}}\left(kâˆˆ{\mathbb{N}}_{0};a,bâˆˆ\mathbb{R}\mathrm{âˆ–}\left\{0\right\};\mathrm{Î²}âˆˆ\mathbb{C}\right).\end{array}$

Thus it is clear that ${}_{H}P_{n,\mathrm{Î²}}\left(x,y,z;k,a,b\right){=}_{H}{P}_{n,\mathrm{Î²}}^{\left(1\right)}\left(x,y,z;k,a,b\right)$ and that we have the following observations at once:

• ${}_{H}P_{n,\mathrm{Î»}}\left(x,y,z;1,1,1\right){=}_{H}{\mathcal{B}}_{n}\left(x,y,z;\mathrm{Î»}\right)$ are the Hermite-based Apostol-Bernoulli polynomials.

• ${}_{H}P_{n,\mathrm{Î»}}\left(x,y,z;0,âˆ’1,1\right){=}_{H}\mathcal{E}\left(x,y,z;\mathrm{Î»}\right)$ are the Hermite-based Apostol-Euler polynomials.

• ${}_{H}P_{n,\frac{\mathrm{Î»}}{2}}\left(x,y,z;1,\frac{âˆ’1}{2},1\right){=}_{H}{\mathcal{G}}_{n}\left(x,y,z;\mathrm{Î»}\right)$ are the Hermite-based Apostol-Genocchi polynomials.

For the other generalization, we refer [11â€“25] and [26]. Now we give some relations between the above mentioned Apostol polynomials.

Using (2.1), we get the following identity at once.

Theorem 2.1 Let $\mathrm{Î±},kâˆˆ{\mathbb{N}}_{0}$; $a,bâˆˆ\mathbb{R}\mathrm{âˆ–}\left\{0\right\}$; $\mathrm{Î²}âˆˆ\mathbb{C}$ be such that the conditions (i)-(iii) are satisfied. Then, the following relation

$\underset{r=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{r}\right)}_{H}{P}_{nâˆ’r,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;k,a,b\right)}_{H}{P}_{r,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(u,v,w;k,a,b\right){=}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x+u,y+v,z+w;k,a,b\right)$

holds true.

Corollary 2.2 For each $nâˆˆ\mathbb{N}$, the following relation

$\underset{k=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{k}\right)}_{H}{\mathcal{B}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{B}}_{k}^{\left(\mathrm{Î²}\right)}\left(u,v,w;\mathrm{Î»}\right){=}_{H}{\mathcal{B}}_{n}^{\left(\mathrm{Î±}+\mathrm{Î²}\right)}\left(x+u,y+v,z+w;\mathrm{Î»}\right)$

holds true for the Hermite-based generalized Apostol-Bernoulli polynomials.

Corollary 2.3 For each $nâˆˆ\mathbb{N}$, the following relation

$\underset{k=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{k}\right)}_{H}{\mathcal{E}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{E}}_{k}^{\left(\mathrm{Î²}\right)}\left(u,v,w;\mathrm{Î»}\right){=}_{H}{\mathcal{E}}_{n}^{\left(\mathrm{Î±}+\mathrm{Î²}\right)}\left(x+u,y+v,z+w;\mathrm{Î»}\right)$

holds true for the Hermite-based generalized Apostol-Euler polynomials.

Corollary 2.4 For each $nâˆˆ\mathbb{N}$, the following relation

$\underset{k=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{k}\right)}_{H}{\mathcal{G}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{G}}_{k}^{\left(\mathrm{Î²}\right)}\left(u,v,w;\mathrm{Î»}\right){=}_{H}{\mathcal{G}}_{n}^{\left(\mathrm{Î±}+\mathrm{Î²}\right)}\left(x+u,y+v,z+w;\mathrm{Î»}\right)$

holds true for the Hermite-based generalized Apostol-Genocchi polynomials.

Theorem 2.5 For each $nâˆˆ\mathbb{N}$, the following relation

$\underset{k=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{k}\right)}_{H}{\mathcal{B}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{E}}_{k}^{\left(\mathrm{Î±}\right)}\left(u,v,w;\mathrm{Î»}\right)={2}_{H}^{n}{\mathcal{B}}_{n}^{\left(\mathrm{Î±}\right)}\left(\frac{x+u}{2},\frac{y+v}{4},\frac{z+w}{8};{\mathrm{Î»}}^{2}\right)$

holds true between the Hermite-based generalized Apostol-Bernoulli and Euler polynomials.

Proof By direct calculations, we have

$\begin{array}{r}{\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{\mathcal{B}}_{n}^{\left(\mathrm{Î±}\right)}\left(\frac{x+u}{2},\frac{y+v}{4},\frac{z+w}{8};{\mathrm{Î»}}^{2}\right)\frac{{\left(2t\right)}^{n}}{n!}\\ \phantom{\rule{1em}{0ex}}={\left(\frac{2t}{{\mathrm{Î»}}^{2}{e}^{2t}âˆ’1}\right)}^{\mathrm{Î±}}exp\left[\left(\frac{x+u}{2}\right)2t+\left(\frac{y+v}{4}\right){\left(2t\right)}^{2}+\left(\frac{z+w}{8}\right){\left(2t\right)}^{3}\right]\\ \phantom{\rule{1em}{0ex}}={\left(\frac{t}{\mathrm{Î»}{e}^{t}âˆ’1}\right)}^{\mathrm{Î±}}exp\left(xt+y{t}^{2}+z{t}^{3}\right){\left(\frac{2}{\mathrm{Î»}{e}^{t}+1}\right)}^{\mathrm{Î±}}exp\left(ut+v{t}^{2}+w{t}^{3}\right)\\ \phantom{\rule{1em}{0ex}}={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{\mathcal{B}}_{n}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right)\frac{{t}^{n}}{n!}{\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{\mathcal{E}}_{k}^{\left(\mathrm{Î±}\right)}\left(u,v,w;\mathrm{Î»}\right)\frac{{t}^{k}}{k!}\\ \phantom{\rule{1em}{0ex}}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\underset{k=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{k}\right)}_{H}{\mathcal{B}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{E}}_{k}^{\left(\mathrm{Î±}\right)}\left(u,v,w;\mathrm{Î»}\right)\frac{{t}^{n}}{n!}.\end{array}$

Comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides, we get the result.â€ƒâ–¡

## 3 Symmetry identities for the unified family

For each $kâˆˆ{\mathbb{N}}_{0}$, the sum ${S}_{k}\left(n\right)={âˆ‘}_{i=0}^{n}{i}^{k}$ is known as the power sum and we have the following generating relation:

$\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{S}_{k}\left(n\right)\frac{{t}^{k}}{k!}=1+{e}^{t}+{e}^{2t}+â‹¯+{e}^{nt}=\frac{{e}^{\left(n+1\right)t}âˆ’1}{{e}^{t}âˆ’1}.$

For an arbitrary real or complex Î», the generalized sum of integer powers ${S}_{k}\left(n,\mathrm{Î»}\right)$ is defined, in [27], via the following generating relation:

$\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{S}_{k}\left(n,\mathrm{Î»}\right)\frac{{t}^{k}}{k!}=\frac{\mathrm{Î»}{e}^{\left(n+1\right)t}âˆ’1}{\mathrm{Î»}{e}^{t}âˆ’1}.$

It clear that ${S}_{k}\left(n,1\right)={S}_{k}\left(n\right)$.

For each $kâˆˆ{\mathbb{N}}_{0}$, the sum ${M}_{k}\left(n\right)={âˆ‘}_{i=0}^{n}{\left(âˆ’1\right)}^{k}{i}^{k}$ is known as the sum of alternative integer powers. The following generating relation is straightforward:

$\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{M}_{k}\left(n\right)\frac{{t}^{k}}{k!}=1âˆ’{e}^{t}+{e}^{2t}âˆ’â‹¯+{\left(âˆ’1\right)}^{n}{e}^{nt}=\frac{1âˆ’{\left(âˆ’{e}^{t}\right)}^{\left(n+1\right)}}{{e}^{t}+1}.$

For an arbitrary real or complex Î», the generalized sum of alternative integer powers ${M}_{k}\left(n,\mathrm{Î»}\right)$ is defined, in [27], by

$\underset{k=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{M}_{k}\left(n,\mathrm{Î»}\right)\frac{{t}^{k}}{k!}=\frac{1âˆ’\mathrm{Î»}{\left(âˆ’{e}^{t}\right)}^{\left(n+1\right)}}{\mathrm{Î»}{e}^{t}+1}.$

Clearly ${M}_{k}\left(n,1\right)={M}_{k}\left(n\right)$. On the other hand, if n is even, then

${S}_{k}\left(n,âˆ’\mathrm{Î»}\right)={M}_{k}\left(n,\mathrm{Î»}\right).$
(3.1)

We start by obtaining certain symmetry identities, which includes the results given in [28â€“32] and [27], when $y=z=0$.

Theorem 3.1 Let $c,d,mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$ be such that the conditions (i)-(iii) are satisfied with t replaced by ct and dt. Then we have the following symmetry identity:

$\begin{array}{r}\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){c}^{nâˆ’r}d^{r+k}{}_{H}{P}_{nâˆ’r,\mathrm{Î²}}^{\left(m\right)}\left(dx,{d}^{2}y,{d}^{3}z;k,a,b\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){S}_{l}{\left(câˆ’1;{\left(\frac{\mathrm{Î²}}{a}\right)}^{b}\right)}_{H}{P}_{râˆ’l,\mathrm{Î²}}^{\left(mâˆ’1\right)}\left(cX,{c}^{2}Y,{c}^{3}Z;k,a,b\right)\\ \phantom{\rule{1em}{0ex}}=\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){d}^{nâˆ’r}c^{r+k}{}_{H}{P}_{nâˆ’r,\mathrm{Î²}}^{\left(m\right)}\left(cx,{c}^{2}y,{c}^{3}z;k,a,b\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){S}_{l}{\left(dâˆ’1;{\left(\frac{\mathrm{Î²}}{a}\right)}^{b}\right)}_{H}{P}_{râˆ’l,\mathrm{Î²}}^{\left(mâˆ’1\right)}\left(dX,{d}^{2}Y,{d}^{3}Z;k,a,b\right).\end{array}$

Proof Let

$G\left(t\right):=\frac{{2}^{\left(1âˆ’k\right)\left(2mâˆ’1\right)}{t}^{2kmâˆ’k}{e}^{cdxt+y{\left(cdt\right)}^{2}+z{\left(cdt\right)}^{3}}\left({\mathrm{Î²}}^{b}{e}^{cdt}âˆ’{a}^{b}\right){e}^{cdXt+Y{\left(cdt\right)}^{2}+Z{\left(cdt\right)}^{3}}}{{\left({\mathrm{Î²}}^{b}{e}^{ct}âˆ’{a}^{b}\right)}^{m}{\left({\mathrm{Î²}}^{b}{e}^{dt}âˆ’{a}^{b}\right)}^{m}}.$

Expanding $G\left(t\right)$ into a series, we get

$\begin{array}{rl}G\left(t\right)=& \frac{1}{{c}^{km}{d}^{k\left(mâˆ’1\right)}}{\left(\frac{{2}^{1âˆ’k}{c}^{k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{ct}âˆ’{a}^{b}}\right)}^{m}{e}^{cdxt+y{\left(cdt\right)}^{2}+z{\left(cdt\right)}^{3}}\left(\frac{{\mathrm{Î²}}^{b}{e}^{cdt}âˆ’{a}^{b}}{{\mathrm{Î²}}^{b}{e}^{dt}âˆ’{a}^{b}}\right)\\ Ã—{\left(\frac{{2}^{1âˆ’k}{d}^{k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{dt}âˆ’{a}^{b}}\right)}^{mâˆ’1}{e}^{cdXt+Y{\left(cdt\right)}^{2}+Z{\left(cdt\right)}^{3}}\\ =& \frac{1}{{c}^{km}{d}^{k\left(mâˆ’1\right)}}\left[{\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}^{\left(m\right)}\left(dx,{d}^{2}y,{d}^{3}z;k,a,b\right)\frac{{\left(ct\right)}^{n}}{n!}\right]\left[\underset{l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{S}_{l}\left(câˆ’1;{\left(\frac{\mathrm{Î²}}{a}\right)}^{b}\right)\frac{{\left(dt\right)}^{l}}{l!}\right]\\ Ã—\left[{\underset{r=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{r,\mathrm{Î²}}^{\left(mâˆ’1\right)}\left(cX,{c}^{2}Y,{c}^{3}Z;k,a,b\right)\frac{{\left(dt\right)}^{r}}{r!}\right].\end{array}$

Now, using Corollary 2 in [[33], p.890], we get

$\begin{array}{rl}G\left(t\right)=& \frac{1}{{c}^{km}{d}^{km}}\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\left[\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){c}^{nâˆ’r}d^{r+k}{}_{H}{P}_{nâˆ’r,\mathrm{Î²}}^{\left(m\right)}\left(dx,{d}^{2}y,{d}^{3}z;k,a,b\right)\\ Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){S}_{l}{\left(câˆ’1;{\left(\frac{\mathrm{Î²}}{a}\right)}^{b}\right)}_{H}{P}_{râˆ’l,\mathrm{Î²}}^{\left(mâˆ’1\right)}\left(cX,{c}^{2}Y,{c}^{3}Z;k,a,b\right)\right]\frac{{t}^{n}}{n!}.\end{array}$
(3.2)

In a similar manner,

$\begin{array}{rl}G\left(t\right)=& \frac{1}{{d}^{km}{c}^{k\left(mâˆ’1\right)}}{\left(\frac{{2}^{1âˆ’k}{d}^{k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{ct}âˆ’{a}^{b}}\right)}^{m}{e}^{cdxt+y{\left(cdt\right)}^{2}+z{\left(cdt\right)}^{3}}\left(\frac{{\mathrm{Î²}}^{b}{e}^{cdt}âˆ’{a}^{b}}{{\mathrm{Î²}}^{b}{e}^{dt}âˆ’{a}^{b}}\right)\\ Ã—{\left(\frac{{2}^{1âˆ’k}{c}^{k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{dt}âˆ’{a}^{b}}\right)}^{mâˆ’1}{e}^{cdXt+Y{\left(cdt\right)}^{2}+Z{\left(cdt\right)}^{3}}\\ =& \frac{1}{{c}^{km}{d}^{km}}\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\left[\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){d}^{nâˆ’r}c^{r+k}{}_{H}{P}_{nâˆ’r,\mathrm{Î²}}^{\left(m\right)}\left(cx,{c}^{2}y,{c}^{3}z;k,a,b\right)\\ Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){S}_{l}{\left(dâˆ’1;{\left(\frac{\mathrm{Î²}}{a}\right)}^{b}\right)}_{H}{P}_{râˆ’l,\mathrm{Î²}}^{\left(mâˆ’1\right)}\left(dX,{d}^{2}Y,{d}^{3}Z;k,a,b\right)\right]\frac{{t}^{n}}{n!}.\end{array}$
(3.3)

From (3.2) and (3.3), we get the result.â€ƒâ–¡

For $k=a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$ we get the following corollary at once.

Corollary 3.2 For all $c,d,mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following symmetry identity for the Hermite based generalized Apostol-Bernoulli polynomials:

$\begin{array}{r}\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){c}^{nâˆ’r}d^{r+1}{}_{H}{\mathcal{B}}_{nâˆ’r}^{\left(m\right)}\left(dx,{d}^{2}y,{d}^{3}z,\mathrm{Î»}\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){S}_{l}{\left(câˆ’1;\mathrm{Î»}\right)}_{H}{\mathcal{B}}_{râˆ’l}^{\left(mâˆ’1\right)}\left(cX,{c}^{2}Y,{c}^{3}Z,\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){d}^{nâˆ’r}c^{r+1}{}_{H}{\mathcal{B}}_{nâˆ’r}^{\left(m\right)}\left(cx,{c}^{2}y,{c}^{3}z,\mathrm{Î»}\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){S}_{l}{\left(dâˆ’1;\mathrm{Î»}\right)}_{H}{\mathcal{B}}_{râˆ’l}^{\left(mâˆ’1\right)}\left(dX,{d}^{2}Y,{d}^{3}Z,\mathrm{Î»}\right).\end{array}$

For $k+1=âˆ’a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$ we get, by considering (3.1) that

Corollary 3.3 For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d,

$\begin{array}{r}\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){c}^{nâˆ’r}d^{r+1}{}_{H}{\mathcal{E}}_{nâˆ’r}^{\left(m\right)}\left(dx,{d}^{2}y,{d}^{3}z,\mathrm{Î»}\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){M}_{l}{\left(câˆ’1;\mathrm{Î»}\right)}_{H}{\mathcal{E}}_{râˆ’l}^{\left(mâˆ’1\right)}\left(cX,{c}^{2}Y,{c}^{3}Z,\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){d}^{nâˆ’r}c^{r+1}{}_{H}{\mathcal{E}}_{nâˆ’r}^{\left(m\right)}\left(cx,{c}^{2}y,{c}^{3}z,\mathrm{Î»}\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){M}_{l}{\left(dâˆ’1;\mathrm{Î»}\right)}_{H}{\mathcal{E}}_{râˆ’l}^{\left(mâˆ’1\right)}\left(dX,{d}^{2}Y,{d}^{3}Z,\mathrm{Î»}\right).\end{array}$

Letting $k=âˆ’2a=b=1$ and $2\mathrm{Î²}=\mathrm{Î»}$ and taking into account (3.1) that we have the following.

Corollary 3.4 For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d, that

$\begin{array}{r}\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){c}^{nâˆ’r}d^{r+1}{}_{H}{\mathcal{G}}_{nâˆ’r}^{\left(m\right)}\left(dx,{d}^{2}y,{d}^{3}z,\mathrm{Î»}\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){M}_{l}{\left(câˆ’1;\mathrm{Î»}\right)}_{H}{\mathcal{G}}_{râˆ’l}^{\left(mâˆ’1\right)}\left(cX,{c}^{2}Y,{c}^{3}Z,\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{r=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right){d}^{nâˆ’r}c^{r+1}{}_{H}{\mathcal{G}}_{nâˆ’r}^{\left(m\right)}\left(cx,{c}^{2}y,{c}^{3}z,\mathrm{Î»}\right)\\ \phantom{\rule{2em}{0ex}}Ã—\underset{l=0}{\overset{r}{âˆ‘}}\left(\genfrac{}{}{0}{}{r}{l}\right){M}_{l}{\left(dâˆ’1;\mathrm{Î»}\right)}_{H}{\mathcal{G}}_{râˆ’l}^{\left(mâˆ’1\right)}\left(dX,{d}^{2}Y,{d}^{3}Z,\mathrm{Î»}\right).\end{array}$

## 4 Closed-form formulae for Hermite-based generalized Apostol polynomials

In this section, taking into account the relations

$\begin{array}{c}{f}_{a,b}^{\left(\mathrm{Î±}\right)}\left(x,y,z;t;k,\mathrm{Î²}\right):={\left(\frac{{2}^{1âˆ’k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{t}âˆ’{a}^{b}}\right)}^{\mathrm{Î±}}{e}^{xt+y{t}^{2}+z{t}^{3}}={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!},\hfill \\ {f}_{a,b}^{\left(1\right)}\left(x,y,z;t;k,\mathrm{Î²}\right):=\left(\frac{{2}^{1âˆ’k}{t}^{k}}{{\mathrm{Î²}}^{b}{e}^{t}âˆ’{a}^{b}}\right){e}^{xt+y{t}^{2}+z{t}^{3}}={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!},\hfill \end{array}$

we observe the following fact:

${\left[{f}_{a,b}^{\left(1\right)}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};t;k,\mathrm{Î²}\right)\right]}^{\mathrm{Î±}}={f}_{a,b}^{\left(\mathrm{Î±}\right)}\left(x,y,z;t;k,\mathrm{Î²}\right).$
(4.1)

Using (4.1), we start by proving the following closed form summation formula:

Theorem 4.1 Let the conditions (i)-(iii) be satisfied. The following summation formula:

$\begin{array}{r}\underset{l=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{l}\right){\left[}_{H}{P}_{nâˆ’l+1,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;k,a,b\right)}_{H}{P}_{l,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\\ \phantom{\rule{1em}{0ex}}âˆ’{\mathrm{Î±}}_{H}{P}_{nâˆ’l,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;k,a,b\right)}_{H}{P}_{l+1,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\right]=0\end{array}$

holds true.

Proof Taking logarithms on both sides of (4.1) and then differentiating with respect to t, we get

$\begin{array}{r}\frac{\mathrm{âˆ‚}{f}_{a,b}^{\left(\mathrm{Î±}\right)}\left(x,y,z;t;k,\mathrm{Î²}\right)}{\mathrm{âˆ‚}t}{f}_{a,b}^{\left(1\right)}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};t;k,\mathrm{Î²}\right)\\ \phantom{\rule{1em}{0ex}}=\mathrm{Î±}{f}_{a,b}^{\left(\mathrm{Î±}\right)}\left(x,y,z;t;k,\mathrm{Î²}\right)\frac{\mathrm{âˆ‚}{f}_{a,b}^{\left(1\right)}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};t;k,\mathrm{Î²}\right)}{\mathrm{âˆ‚}t}.\end{array}$

Inserting the corresponding generating relations, we obtain

$\begin{array}{r}\underset{n=1}{\overset{\mathrm{âˆž}}{âˆ‘}}{n}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{nâˆ’1}}{n!}{\underset{l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{l,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\frac{{t}^{l}}{l!}\\ \phantom{\rule{1em}{0ex}}=\mathrm{Î±}{\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\underset{l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{l}_{H}{P}_{l,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\frac{{t}^{lâˆ’1}}{l!},\end{array}$

and hence

$\begin{array}{r}{\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n+1,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}{\underset{l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{l,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\frac{{t}^{l}}{l!}\\ \phantom{\rule{1em}{0ex}}=\mathrm{Î±}{\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}{\underset{l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{l+1,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\frac{{t}^{l}}{l!}.\end{array}$

Using the fact that (see [[34], p.101, Lemma 3])

$\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\underset{l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}A\left(n,l\right)=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\underset{l=0}{\overset{n}{âˆ‘}}A\left(nâˆ’l,l\right),$
(4.2)

we get

$\begin{array}{r}\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\left[\underset{l=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{l}\right)}_{H}{P}_{nâˆ’l+1,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;k,a,b\right)}_{H}{P}_{l,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\right]\frac{{t}^{n}}{n!}\\ \phantom{\rule{1em}{0ex}}=\mathrm{Î±}\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\left[\underset{l=0}{\overset{n}{âˆ‘}}{\left(\genfrac{}{}{0}{}{n}{l}\right)}_{H}{P}_{nâˆ’l,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;k,a,b\right)}_{H}{P}_{l+1,\mathrm{Î²}}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};k,a,b\right)\right]\frac{{t}^{n}}{n!}.\end{array}$

Whence the result.â€ƒâ–¡

Corollary 4.2 Let $k=a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$. For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following closed form summation formula for the generalized Apostol-Bernoulli polynomials:

$\begin{array}{r}\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[{}_{H}\mathcal{B}_{nâˆ’k+1}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{B}}_{k}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}âˆ’{\mathrm{Î±}}_{H}{\mathcal{B}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right){\mathcal{B}}_{k+1}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};\mathrm{Î»}\right)\right]=0.\end{array}$

Corollary 4.3 Let $k+1=âˆ’a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$. For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following closed form summation formula for the generalized Apostol-Euler polynomials:

$\begin{array}{r}\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[{}_{H}\mathcal{E}_{nâˆ’k+1}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{E}}_{k}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}âˆ’{\mathrm{Î±}}_{H}{\mathcal{E}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right){\mathcal{E}}_{k+1}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};\mathrm{Î»}\right)\right]=0.\end{array}$

Corollary 4.4 Let $k=âˆ’2a=b=1$ and $2\mathrm{Î²}=\mathrm{Î»}$. For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following closed form summation formula for the generalized Apostol-Genocchi polynomials:

$\begin{array}{r}\underset{k=0}{\overset{n}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left[{}_{H}\mathcal{G}_{nâˆ’k+1}^{\left(\mathrm{Î±}\right)}{\left(x,y,z;\mathrm{Î»}\right)}_{H}{\mathcal{G}}_{k}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}âˆ’{\mathrm{Î±}}_{H}{\mathcal{G}}_{nâˆ’k}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right){\mathcal{G}}_{k+1}\left(\frac{x}{\mathrm{Î±}},\frac{y}{\mathrm{Î±}},\frac{z}{\mathrm{Î±}};\mathrm{Î»}\right)\right]=0.\end{array}$

Theorem 4.5 Let the conditions (i)-(iii) be satisfied. Then we have the following relation between Hermite based Apostol polynomials and 3d-Hermite polynomials:

$\begin{array}{r}{}_{H}P_{n+m,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(X,Y,Z;k,a,b\right)\\ \phantom{\rule{1em}{0ex}}=\underset{r,l=0}{\overset{n,m}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right)\left(\genfrac{}{}{0}{}{m}{l}\right){H}_{r+l}^{\left(3\right)}{\left(Xâˆ’x,Yâˆ’y,Zâˆ’z\right)}_{H}{P}_{n+mâˆ’râˆ’l}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right).\end{array}$

Proof From (2.1), we can write that

$\begin{array}{rl}{\left(\frac{{2}^{1âˆ’k}{\left(t+w\right)}^{k}}{{\mathrm{Î²}}^{b}{e}^{t+w}âˆ’{a}^{b}}\right)}^{\mathrm{Î±}}{e}^{x\left(t+w\right)+y{\left(t+w\right)}^{2}+z{\left(t+w\right)}^{3}}& ={\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{\left(t+w\right)}^{n}}{n!}\\ ={\underset{n,m=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n+m,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\frac{{w}^{m}}{m!}.\end{array}$
(4.3)

Therefore, we get

${\left(\frac{{2}^{1âˆ’k}{\left(t+w\right)}^{k}}{{\mathrm{Î²}}^{b}{e}^{t+w}âˆ’{a}^{b}}\right)}^{\mathrm{Î±}}={e}^{âˆ’x\left(t+w\right)âˆ’y{\left(t+w\right)}^{2}âˆ’z{\left(t+w\right)}^{3}}{\underset{n,m=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n+m,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\frac{{w}^{m}}{m!}.$

Multiplying both sides by ${e}^{X\left(t+w\right)+Y{\left(t+w\right)}^{2}+Z{\left(t+w\right)}^{3}}$, we have

$\begin{array}{r}{\left(\frac{{2}^{1âˆ’k}{\left(t+w\right)}^{k}}{{\mathrm{Î²}}^{b}{e}^{t+w}âˆ’{a}^{b}}\right)}^{\mathrm{Î±}}{e}^{X\left(t+w\right)+Y{\left(t+w\right)}^{2}+Z{\left(t+w\right)}^{3}}\\ \phantom{\rule{1em}{0ex}}={e}^{\left(Xâˆ’x\right)\left(t+w\right)+\left(Yâˆ’y\right){\left(t+w\right)}^{2}+\left(Zâˆ’z\right){\left(t+w\right)}^{3}}{\underset{n,m=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n+m,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\frac{{w}^{m}}{m!}.\end{array}$

Taking into account (1.1) and (4.3), then using (4.2), we get

$\begin{array}{r}{\underset{n,m=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n+m,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(X,Y,Z;k,a,b\right)\frac{{t}^{n}}{n!}\frac{{w}^{m}}{m!}\\ \phantom{\rule{1em}{0ex}}={\underset{n,m=0}{\overset{\mathrm{âˆž}}{âˆ‘}}}_{H}{P}_{n+m,\mathrm{Î²}}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\frac{{w}^{m}}{m!}\underset{r,l=0}{\overset{\mathrm{âˆž}}{âˆ‘}}{H}_{r+l}^{\left(3\right)}\left(Xâˆ’x,Yâˆ’y,Zâˆ’z\right)\frac{{t}^{r}}{r!}\frac{{w}^{l}}{l!}\\ \phantom{\rule{1em}{0ex}}=\underset{n,m=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\underset{r,l=0}{\overset{n,m}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{r}\right)\left(\genfrac{}{}{0}{}{m}{l}\right){H}_{r+l}^{\left(3\right)}{\left(Xâˆ’x,Yâˆ’y,Zâˆ’z\right)}_{H}{P}_{n+mâˆ’râˆ’l}^{\left(\mathrm{Î±}\right)}\left(x,y,z;k,a,b\right)\frac{{t}^{n}}{n!}\frac{{w}^{m}}{m!}.\end{array}$

Whence the result.â€ƒâ–¡

Corollary 4.6 Let $k=a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$. For all $c,d,mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following summation formula between the Hermite-based generalized Apostol-Bernoulli polynomials and 3d-Hermite polynomials:

$\begin{array}{r}{}_{H}\mathcal{B}_{n+m}^{\left(\mathrm{Î±}\right)}\left(X,Y,Z;\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{k,l=0}{\overset{n,m}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{m}{l}\right){H}_{k+l}^{\left(3\right)}{\left(Xâˆ’x,Yâˆ’y,Zâˆ’z\right)}_{H}{\mathcal{B}}_{n+mâˆ’kâˆ’l}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right).\end{array}$

Corollary 4.7 Let $k+1=âˆ’a=b=1$ and $\mathrm{Î²}=\mathrm{Î»}$. For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following summation formula between the Hermite-based generalized Apostol-Euler polynomials and 3d-Hermite polynomials:

$\begin{array}{r}{}_{H}\mathcal{E}_{n+m}^{\left(\mathrm{Î±}\right)}\left(X,Y,Z;\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{k,l=0}{\overset{n,m}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{m}{l}\right){H}_{k+l}^{\left(3\right)}{\left(Xâˆ’x,Yâˆ’y,Zâˆ’z\right)}_{H}{\mathcal{E}}_{n+mâˆ’kâˆ’l}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right).\end{array}$

Corollary 4.8 Let $k=âˆ’2a=b=1$ and $2\mathrm{Î²}=\mathrm{Î»}$. For all $mâˆˆ\mathbb{N}$, $nâˆˆ{\mathbb{N}}_{0}$, $\mathrm{Î»}âˆˆ\mathbb{C}$, we have the following summation formula between the Hermite-based generalized Apostol-Genocchi polynomials and 3d-Hermite polynomials:

$\begin{array}{r}{}_{H}\mathcal{G}_{n+m}^{\left(\mathrm{Î±}\right)}\left(X,Y,Z;\mathrm{Î»}\right)\\ \phantom{\rule{1em}{0ex}}=\underset{k,l=0}{\overset{n,m}{âˆ‘}}\left(\genfrac{}{}{0}{}{n}{k}\right)\left(\genfrac{}{}{0}{}{m}{l}\right){H}_{k+l}^{\left(3\right)}{\left(Xâˆ’x,Yâˆ’y,Zâˆ’z\right)}_{H}{\mathcal{G}}_{n+mâˆ’kâˆ’l}^{\left(\mathrm{Î±}\right)}\left(x,y,z;\mathrm{Î»}\right).\end{array}$

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## Acknowledgements

Dedicated to Professor Hari M Srivastava.

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Correspondence to Mehmet Ali Ã–zarslan.

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The author completed the paper himself. The author read and approved the final manuscript.

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Ã–zarslan, M.A. Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials. Adv Differ Equ 2013, 116 (2013). https://doi.org/10.1186/1687-1847-2013-116