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

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

where

$$\mathcal{M}=x+2y\frac{\partial }{\partial x}+3z\frac{{\partial }^2}{\partial x^2}$$

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

$$exp(xt+y{t}^2+z{t}^3)=\sum _{n=0}^{\mathrm{\infty }}{H}_n^{(3)}(x,y,z)\frac{t^n}{n!}$$

(1.1)

and

$$A(t)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{t}^n,a_0\ne 0.$$

By using the Berry decoupling identity,

$$e^{A+B}=e^{m^2/12}{e}^{((\frac{-m}{2})A^{1/2}+A)}{e}^B,[A,B]=m{A}^{1/2}$$

they obtained the generating function of the Hermite-based Appell polynomials ${}_{H}A_n(x,y,z)$ as

$$\mathcal{G}(x,y,z;t)=A(t)exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{A}_n(x,y,z)\frac{t^n}{n!}.$$

Letting $A(t)=\frac{t}{e^t-1}$, they defined Hermite-Bernoulli polynomials ${}_{H}B_n(x,y,z)$ by

$$\frac{t}{e^t-1}exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{B}_n(x,y,z)\frac{t^n}{n!},|t|<2\pi .$$

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

$$\frac{2}{e^t+1}exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{E}_n(x,y,z)\frac{t^n}{n!},|t|<\pi$$

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

$$\frac{2t}{e^t+1}exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{G}_n(x,y,z)\frac{t^n}{n!},|t|<\pi .$$

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

$$\begin{array}{r}{f}_{a,b}^{(\alpha )}(x;t;k,\beta ):={\left(\frac{2^{1-k}{t}^k}{{\beta }^b{e}^t-a^b}\right)}^{\alpha }{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{P}_{n,\beta }^{(\alpha )}(x;k,a,b)\frac{t^n}{n!}\\ (k\in {\mathbb{N}}_0;a,b\in \mathbb{R}\mathrm{\setminus }\{0\};\alpha ,\beta \in \mathbb{C})\end{array}$$

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 $P_{n,\beta }^{(1)}(x,y,z;k,a,b)$ 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 ${}_{H}P_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)$ 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:

$$\begin{array}{r}{f}_{a,b}^{(\alpha )}(x,y,z;t;k,\beta ):={\left(\frac{2^{1-k}{t}^k}{{\beta }^b{e}^t-a^b}\right)}^{\alpha }{e}^{xt+y{t}^2+z{t}^3}={\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\\ (k\in {\mathbb{N}}_0;a,b\in \mathbb{R}\mathrm{\setminus }\{0\};\alpha ,\beta \in \mathbb{C}).\end{array}$$

(2.1)

For the existence of the expansion, we need

• (i) $|t|<2\pi$ when $\alpha \in \mathbb{C}$, $k=1$ and ${(\frac{\beta }{a})}^b=1$; $|t|<2\pi$ when $\alpha \in {\mathbb{N}}_0$, $k=2,3,\dots$ and ${(\frac{\beta }{a})}^b=1$; $|t|<|blog(\frac{\beta }{a})|$ when $\alpha \in {\mathbb{N}}_0$, $k\in \mathbb{N}$ and ${(\frac{\beta }{a})}^b\ne 1$ (or $\ne -1$); $x,y,z\in \mathbb{R}$, $\beta \in \mathbb{C}$, $a,b\in \mathbb{C}/\{0\}$; $1^{\alpha }:=1$;

• (ii) $|t|<\pi$ when ${(\frac{\beta }{a})}^b=-1$; $|t|<|blog(\frac{\beta }{a})|$ when ${(\frac{\beta }{a})}^b\ne -1$; $x,y,z\in \mathbb{R}$, $k=0$, $\alpha ,\beta \in \mathbb{C}$, $a,b\in \mathbb{C}/\{0\}$; $1^{\alpha }:=1$;

• (iii) $|t|<\pi$ when $\alpha \in {\mathbb{N}}_0$ and ${(\frac{\beta }{a})}^b=-1$; $x,y,z\in \mathbb{R}$, $k\in \mathbb{N}$, $\beta \in \mathbb{C}$, $a,b\in \mathbb{C}/\{0\}$; $1^{\alpha }:=1$,

where $w=|w|e^{i\theta }$, $-\pi \le \theta <\pi$ and $log(w)=log(|w|)+i\theta$.

For $k=a=b=1$ and $\beta =\lambda$ in (2.1), we define the following.

Definition 2.1 Let $\alpha \in {\mathbb{N}}_0$, λ be an arbitrary (real or complex) parameter and $x,y,z\in \mathbb{R}$. The Hermite-based generalized Apostol-Bernoulli polynomials are defined by

$$\begin{array}{r}{\left(\frac{t}{\lambda e^t-1}\right)}^{\alpha }exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{\mathcal{B}}_n^{(\alpha )}(x,y,z;\lambda )\frac{t^n}{n!}\\ (|t|<2\pi \text{ when }\alpha \in \mathbb{C}\text{ and }\lambda =1;|t|<|log(\lambda )|\\ \text{when }\alpha \in {\mathbb{N}}_0\text{ and }\lambda \ne 1;x,y,z\in \mathbb{R};1^{\alpha }:=1).\end{array}$$

It is clear that

$${}_{H}P_{n,\lambda }^{(\alpha )}(x,y,z;1,1,1){=}_H{\mathcal{B}}_n^{(\alpha )}(x,y,z;\lambda ).$$

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

• ${}_H\mathcal{B}_n^{(1)}(x,y,z;\lambda ):{=}_H{\mathcal{B}}_n(x,y,z;\lambda )$ is called Hermite-based Apostol-Bernoulli polynomials.

• ${}_H\mathcal{B}_n(x,y,z;1){=}_H{B}_n(x,y,z)$ is the Hermite-Bernoulli polynomials.

• ${}_H\mathcal{B}_n(x,0,0;\lambda ):={\mathcal{B}}_n(x;\lambda )$ is the Apostol-Bernoulli polynomials (see [4–7]). When $\lambda =1$, we have the classical Bernoulli polynomials.

• ${\mathcal{B}}_n(0;\lambda ):={\mathcal{B}}_n(\lambda )$ are the Apostol-Bernoulli numbers. $\lambda =1$ gives the classical Bernoulli numbers.

Setting $k+1=-a=b=1$ and $\beta =\lambda$ in (2.1), we get the following.

Definition 2.2 Let α and λ ($\ne -1$) be an arbitrary (real or complex) parameter and $x,y,z\in \mathbb{R}$. The Hermite-based generalized Apostol-Euler polynomials are defined by

$$\begin{array}{r}{\left(\frac{2}{\lambda e^t+1}\right)}^{\alpha }exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{\mathcal{E}}_n^{(\alpha )}(x,y,z;\lambda )\frac{t^n}{n!}\\ (|t|<\pi \text{ when }\lambda =1;|t|<|log(-\lambda )|\text{ when }\lambda \ne 1;x,y,z\in \mathbb{R},\alpha \in \mathbb{C};1^{\alpha }:=1).\end{array}$$

Obviously, we have

$${}_{H}P_{n,\lambda }^{(\alpha )}(x,y,z;0,-1,1){=}_H{\mathcal{E}}_n^{(\alpha )}(x,y,z;\lambda ).$$

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

• ${}_H\mathcal{E}_n^{(1)}(x,y,z;\lambda ):{=}_H{\mathcal{E}}_n(x,y,z;\lambda )$ is called Hermite-based Apostol-Euler polynomials.

• ${}_H\mathcal{E}_n(x,y,z;1){=}_H{E}_n(x,y,z)$ is the Hermite-Euler polynomials.

• ${}_H\mathcal{E}_n(x,0,0;\lambda ):={\mathcal{E}}_n(x;\lambda )$ is the Apostol-Euler polynomials (see [8]). For $\lambda =1$, we have the classical Euler polynomials.

• $2^n{\mathcal{E}}_n(\frac{1}{2};\lambda ):={\mathcal{E}}_n(\lambda )$ are the Apostol-Euler numbers. The case $\lambda =1$ gives the classical Euler numbers.

Choosing $k=-2a=b=1$ and $2\beta =\lambda$ in (2.1), we define the following.

Definition 2.3 Let α and λ ($\ne -1$) be an arbitrary (real or complex) parameter and $x,y,z\in \mathbb{R}$. The Hermite-based generalized Apostol-Genocchi polynomials are defined by

$$\begin{array}{r}{\left(\frac{2t}{\lambda e^t+1}\right)}^{\alpha }exp(xt+y{t}^2+z{t}^3)={\sum _{n=0}^{\mathrm{\infty }}}_H{\mathcal{G}}_n^{(\alpha )}(x,y,z;\lambda )\frac{t^n}{n!}\\ (|t|<\pi \text{ when }\alpha \in {\mathbb{N}}_0\text{ and }\lambda =1;|t|<|log(-\lambda )|\\ \text{when }\alpha \in {\mathbb{N}}_0\text{ and }\lambda \ne 1;x,y,z\in \mathbb{R};1^{\alpha }:=1).\end{array}$$

It is easily seen that

$${}_{H}P_{n,\frac{\lambda }{2}}^{(\alpha )}(x,y,z;1,\frac{-1}{2},1){=}_H{\mathcal{G}}_n^{\alpha }(x,y,z;\lambda ).$$

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

• ${}_H\mathcal{G}_n^{(1)}(x,y,z;\lambda ):{=}_H{\mathcal{G}}_n(x,y,z;\lambda )$ is called Hermite-based Apostol-Genocchi polynomials.

• ${}_H\mathcal{G}_n(x,y,z;1){=}_H{G}_n(x,y,z)$ is the Hermite-Genocchi polynomials.

• ${}_H\mathcal{G}_n(x,0,0;\lambda ):={\mathcal{G}}_n(x;\lambda )$ is the Apostol-Genocchi polynomials (see [9, 10]). When $\lambda =1$, we have the classical Genocchi polynomials.

• ${\mathcal{G}}_n(0;\lambda ):={\mathcal{G}}_n(\lambda )$ are the Apostol-Genocchi numbers. $\lambda =1$ gives the classical Genocchi numbers.

Finally we define the unified Hermite-based Apostol polynomials by

$$\begin{array}{r}{f}_{a,b}^{(1)}(x;t;k,\beta ):=\frac{2^{1-k}{t}^k}{{\beta }^b{e}^t-a^b}{e}^{xt+y{t}^2+z{t}^3}={\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }(x,y,z;k,a,b)\frac{t^n}{n!}\\ (k\in {\mathbb{N}}_0;a,b\in \mathbb{R}\mathrm{\setminus }\{0\};\beta \in \mathbb{C}).\end{array}$$

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

• ${}_{H}P_{n,\lambda }(x,y,z;1,1,1){=}_H{\mathcal{B}}_n(x,y,z;\lambda )$ are the Hermite-based Apostol-Bernoulli polynomials.

• ${}_{H}P_{n,\lambda }(x,y,z;0,-1,1){=}_H\mathcal{E}(x,y,z;\lambda )$ are the Hermite-based Apostol-Euler polynomials.

• ${}_{H}P_{n,\frac{\lambda }{2}}(x,y,z;1,\frac{-1}{2},1){=}_H{\mathcal{G}}_n(x,y,z;\lambda )$ are the Hermite-based Apostol-Genocchi polynomials.

For the other generalization, we refer [11–25] 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 $\alpha ,k\in {\mathbb{N}}_0$; $a,b\in \mathbb{R}\mathrm{\setminus }\{0\}$; $\beta \in \mathbb{C}$ be such that the conditions (i)-(iii) are satisfied. Then, the following relation

$$\sum _{r=0}^n{\left(\genfrac{}{}{0ex}{}{n}{r}\right)}_H{P}_{n-r,\beta }^{(\alpha )}{(x,y,z;k,a,b)}_H{P}_{r,\beta }^{(\alpha )}(u,v,w;k,a,b){=}_H{P}_{n,\beta }^{(\alpha )}(x+u,y+v,z+w;k,a,b)$$

holds true.

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

$$\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_H{\mathcal{B}}_{n-k}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{B}}_k^{(\beta )}(u,v,w;\lambda ){=}_H{\mathcal{B}}_n^{(\alpha +\beta )}(x+u,y+v,z+w;\lambda )$$

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

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

$$\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_H{\mathcal{E}}_{n-k}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{E}}_k^{(\beta )}(u,v,w;\lambda ){=}_H{\mathcal{E}}_n^{(\alpha +\beta )}(x+u,y+v,z+w;\lambda )$$

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

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

$$\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_H{\mathcal{G}}_{n-k}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{G}}_k^{(\beta )}(u,v,w;\lambda ){=}_H{\mathcal{G}}_n^{(\alpha +\beta )}(x+u,y+v,z+w;\lambda )$$

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

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

$$\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_H{\mathcal{B}}_{n-k}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{E}}_k^{(\alpha )}(u,v,w;\lambda )=2_H^n{\mathcal{B}}_n^{(\alpha )}(\frac{x+u}{2},\frac{y+v}{4},\frac{z+w}{8};{\lambda }^2)$$

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

Proof By direct calculations, we have

$$\begin{array}{r}{\sum _{n=0}^{\mathrm{\infty }}}_H{\mathcal{B}}_n^{(\alpha )}(\frac{x+u}{2},\frac{y+v}{4},\frac{z+w}{8};{\lambda }^2)\frac{{(2t)}^n}{n!}\\ ={\left(\frac{2t}{{\lambda }^2{e}^{2t}-1}\right)}^{\alpha }exp[\left(\frac{x+u}{2}\right)2t+\left(\frac{y+v}{4}\right){(2t)}^2+\left(\frac{z+w}{8}\right){(2t)}^3]\\ ={\left(\frac{t}{\lambda e^t-1}\right)}^{\alpha }exp(xt+y{t}^2+z{t}^3){\left(\frac{2}{\lambda e^t+1}\right)}^{\alpha }exp(ut+v{t}^2+w{t}^3)\\ ={\sum _{n=0}^{\mathrm{\infty }}}_H{\mathcal{B}}_n^{(\alpha )}(x,y,z;\lambda )\frac{t^n}{n!}{\sum _{k=0}^{\mathrm{\infty }}}_H{\mathcal{E}}_k^{(\alpha )}(u,v,w;\lambda )\frac{t^k}{k!}\\ =\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_H{\mathcal{B}}_{n-k}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{E}}_k^{(\alpha )}(u,v,w;\lambda )\frac{t^n}{n!}.\end{array}$$

Comparing the coefficients of $\frac{t^n}{n!}$ on both sides, we get the result. □

3 Symmetry identities for the unified family

For each $k\in {\mathbb{N}}_0$, the sum $S_k(n)={\sum }_{i=0}^n{i}^k$ is known as the power sum and we have the following generating relation:

$$\sum _{k=0}^{\mathrm{\infty }}{S}_k(n)\frac{t^k}{k!}=1+e^t+e^{2t}+\cdots +e^{nt}=\frac{e^{(n+1)t}-1}{e^t-1}.$$

For an arbitrary real or complex λ, the generalized sum of integer powers $S_k(n,\lambda )$ is defined, in [27], via the following generating relation:

$$\sum _{k=0}^{\mathrm{\infty }}{S}_k(n,\lambda )\frac{t^k}{k!}=\frac{\lambda e^{(n+1)t}-1}{\lambda e^t-1}.$$

It clear that $S_k(n,1)=S_k(n)$.

For each $k\in {\mathbb{N}}_0$, the sum $M_k(n)={\sum }_{i=0}^n{(-1)}^k{i}^k$ is known as the sum of alternative integer powers. The following generating relation is straightforward:

$$\sum _{k=0}^{\mathrm{\infty }}{M}_k(n)\frac{t^k}{k!}=1-e^t+e^{2t}-\cdots +{(-1)}^n{e}^{nt}=\frac{1-{(-e^t)}^{(n+1)}}{e^t+1}.$$

For an arbitrary real or complex λ, the generalized sum of alternative integer powers $M_k(n,\lambda )$ is defined, in [27], by

$$\sum _{k=0}^{\mathrm{\infty }}{M}_k(n,\lambda )\frac{t^k}{k!}=\frac{1-\lambda {(-e^t)}^{(n+1)}}{\lambda e^t+1}.$$

Clearly $M_k(n,1)=M_k(n)$. On the other hand, if n is even, then

$$S_k(n,-\lambda )=M_k(n,\lambda ).$$

(3.1)

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

Theorem 3.1 Let $c,d,m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$ be such that the conditions (i)-(iii) are satisfied with t replaced by ct and dt. Then we have the following symmetry identity:

$$\begin{array}{r}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)c^{n-r}d^{r+k}{}_H{P}_{n-r,\beta }^{(m)}(dx,d^{2}y,d^{3}z;k,a,b)\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)S_l{(c-1;{\left(\frac{\beta }{a}\right)}^b)}_H{P}_{r-l,\beta }^{(m-1)}(cX,c^{2}Y,c^{3}Z;k,a,b)\\ =\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)d^{n-r}c^{r+k}{}_H{P}_{n-r,\beta }^{(m)}(cx,c^{2}y,c^{3}z;k,a,b)\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)S_l{(d-1;{\left(\frac{\beta }{a}\right)}^b)}_H{P}_{r-l,\beta }^{(m-1)}(dX,d^{2}Y,d^{3}Z;k,a,b).\end{array}$$

Proof Let

$$G(t):=\frac{2^{(1-k)(2m-1)}{t}^{2km-k}{e}^{cdxt+y{(cdt)}^2+z{(cdt)}^3}({\beta }^b{e}^{cdt}-a^b)e^{cdXt+Y{(cdt)}^2+Z{(cdt)}^3}}{{({\beta }^b{e}^{ct}-a^b)}^m{({\beta }^b{e}^{dt}-a^b)}^m}.$$

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

$$\begin{array}{rl}G(t)=& \frac{1}{c^{km}{d}^{k(m-1)}}{\left(\frac{2^{1-k}{c}^k{t}^k}{{\beta }^b{e}^{ct}-a^b}\right)}^m{e}^{cdxt+y{(cdt)}^2+z{(cdt)}^3}\left(\frac{{\beta }^b{e}^{cdt}-a^b}{{\beta }^b{e}^{dt}-a^b}\right)\\ \times {\left(\frac{2^{1-k}{d}^k{t}^k}{{\beta }^b{e}^{dt}-a^b}\right)}^{m-1}{e}^{cdXt+Y{(cdt)}^2+Z{(cdt)}^3}\\ =& \frac{1}{c^{km}{d}^{k(m-1)}}\left[{\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }^{(m)}(dx,d^{2}y,d^{3}z;k,a,b)\frac{{(ct)}^n}{n!}\right]\left[\sum _{l=0}^{\mathrm{\infty }}{S}_l(c-1;{\left(\frac{\beta }{a}\right)}^b)\frac{{(dt)}^l}{l!}\right]\\ \times \left[{\sum _{r=0}^{\mathrm{\infty }}}_H{P}_{r,\beta }^{(m-1)}(cX,c^{2}Y,c^{3}Z;k,a,b)\frac{{(dt)}^r}{r!}\right].\end{array}$$

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

$$\begin{array}{rl}G(t)=& \frac{1}{c^{km}{d}^{km}}\sum _{n=0}^{\mathrm{\infty }}[\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)c^{n-r}d^{r+k}{}_H{P}_{n-r,\beta }^{(m)}(dx,d^{2}y,d^{3}z;k,a,b)\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)S_l{(c-1;{\left(\frac{\beta }{a}\right)}^b)}_H{P}_{r-l,\beta }^{(m-1)}(cX,c^{2}Y,c^{3}Z;k,a,b)]\frac{t^n}{n!}.\end{array}$$

(3.2)

In a similar manner,

$$\begin{array}{rl}G(t)=& \frac{1}{d^{km}{c}^{k(m-1)}}{\left(\frac{2^{1-k}{d}^k{t}^k}{{\beta }^b{e}^{ct}-a^b}\right)}^m{e}^{cdxt+y{(cdt)}^2+z{(cdt)}^3}\left(\frac{{\beta }^b{e}^{cdt}-a^b}{{\beta }^b{e}^{dt}-a^b}\right)\\ \times {\left(\frac{2^{1-k}{c}^k{t}^k}{{\beta }^b{e}^{dt}-a^b}\right)}^{m-1}{e}^{cdXt+Y{(cdt)}^2+Z{(cdt)}^3}\\ =& \frac{1}{c^{km}{d}^{km}}\sum _{n=0}^{\mathrm{\infty }}[\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)d^{n-r}c^{r+k}{}_H{P}_{n-r,\beta }^{(m)}(cx,c^{2}y,c^{3}z;k,a,b)\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)S_l{(d-1;{\left(\frac{\beta }{a}\right)}^b)}_H{P}_{r-l,\beta }^{(m-1)}(dX,d^{2}Y,d^{3}Z;k,a,b)]\frac{t^n}{n!}.\end{array}$$

(3.3)

From (3.2) and (3.3), we get the result. □

For $k=a=b=1$ and $\beta =\lambda$ we get the following corollary at once.

Corollary 3.2 For all $c,d,m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following symmetry identity for the Hermite based generalized Apostol-Bernoulli polynomials:

$$\begin{array}{r}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)c^{n-r}d^{r+1}{}_H{\mathcal{B}}_{n-r}^{(m)}(dx,d^{2}y,d^{3}z,\lambda )\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)S_l{(c-1;\lambda )}_H{\mathcal{B}}_{r-l}^{(m-1)}(cX,c^{2}Y,c^{3}Z,\lambda )\\ =\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)d^{n-r}c^{r+1}{}_H{\mathcal{B}}_{n-r}^{(m)}(cx,c^{2}y,c^{3}z,\lambda )\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)S_l{(d-1;\lambda )}_H{\mathcal{B}}_{r-l}^{(m-1)}(dX,d^{2}Y,d^{3}Z,\lambda ).\end{array}$$

For $k+1=-a=b=1$ and $\beta =\lambda$ we get, by considering (3.1) that

Corollary 3.3 For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d,

$$\begin{array}{r}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)c^{n-r}d^{r+1}{}_H{\mathcal{E}}_{n-r}^{(m)}(dx,d^{2}y,d^{3}z,\lambda )\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)M_l{(c-1;\lambda )}_H{\mathcal{E}}_{r-l}^{(m-1)}(cX,c^{2}Y,c^{3}Z,\lambda )\\ =\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)d^{n-r}c^{r+1}{}_H{\mathcal{E}}_{n-r}^{(m)}(cx,c^{2}y,c^{3}z,\lambda )\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)M_l{(d-1;\lambda )}_H{\mathcal{E}}_{r-l}^{(m-1)}(dX,d^{2}Y,d^{3}Z,\lambda ).\end{array}$$

Letting $k=-2a=b=1$ and $2\beta =\lambda$ and taking into account (3.1) that we have the following.

Corollary 3.4 For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d, that

$$\begin{array}{r}\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)c^{n-r}d^{r+1}{}_H{\mathcal{G}}_{n-r}^{(m)}(dx,d^{2}y,d^{3}z,\lambda )\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)M_l{(c-1;\lambda )}_H{\mathcal{G}}_{r-l}^{(m-1)}(cX,c^{2}Y,c^{3}Z,\lambda )\\ =\sum _{r=0}^n\left(\genfrac{}{}{0ex}{}{n}{r}\right)d^{n-r}c^{r+1}{}_H{\mathcal{G}}_{n-r}^{(m)}(cx,c^{2}y,c^{3}z,\lambda )\\ \times \sum _{l=0}^r\left(\genfrac{}{}{0ex}{}{r}{l}\right)M_l{(d-1;\lambda )}_H{\mathcal{G}}_{r-l}^{(m-1)}(dX,d^{2}Y,d^{3}Z,\lambda ).\end{array}$$

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

In this section, taking into account the relations

$$\begin{array}{c}{f}_{a,b}^{(\alpha )}(x,y,z;t;k,\beta ):={\left(\frac{2^{1-k}{t}^k}{{\beta }^b{e}^t-a^b}\right)}^{\alpha }{e}^{xt+y{t}^2+z{t}^3}={\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!},\hfill \\ f_{a,b}^{(1)}(x,y,z;t;k,\beta ):=\left(\frac{2^{1-k}{t}^k}{{\beta }^b{e}^t-a^b}\right)e^{xt+y{t}^2+z{t}^3}={\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }(x,y,z;k,a,b)\frac{t^n}{n!},\hfill \end{array}$$

we observe the following fact:

$${\left[f_{a,b}^{(1)}(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };t;k,\beta )\right]}^{\alpha }=f_{a,b}^{(\alpha )}(x,y,z;t;k,\beta ).$$

(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:

$$\begin{array}{r}\sum _{l=0}^n\left(\genfrac{}{}{0ex}{}{n}{l}\right){[}_H{P}_{n-l+1,\beta }^{(\alpha )}{(x,y,z;k,a,b)}_H{P}_{l,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\\ -{\alpha }_H{P}_{n-l,\beta }^{(\alpha )}{(x,y,z;k,a,b)}_H{P}_{l+1,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)]=0\end{array}$$

holds true.

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

$$\begin{array}{r}\frac{\partial f_{a,b}^{(\alpha )}(x,y,z;t;k,\beta )}{\partial t}{f}_{a,b}^{(1)}(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };t;k,\beta )\\ =\alpha f_{a,b}^{(\alpha )}(x,y,z;t;k,\beta )\frac{\partial f_{a,b}^{(1)}(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };t;k,\beta )}{\partial t}.\end{array}$$

Inserting the corresponding generating relations, we obtain

$$\begin{array}{r}\sum _{n=1}^{\mathrm{\infty }}{n}_H{P}_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^{n-1}}{n!}{\sum _{l=0}^{\mathrm{\infty }}}_H{P}_{l,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\frac{t^l}{l!}\\ =\alpha {\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\sum _{l=0}^{\mathrm{\infty }}{l}_H{P}_{l,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\frac{t^{l-1}}{l!},\end{array}$$

and hence

$$\begin{array}{r}{\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n+1,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}{\sum _{l=0}^{\mathrm{\infty }}}_H{P}_{l,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\frac{t^l}{l!}\\ =\alpha {\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}{\sum _{l=0}^{\mathrm{\infty }}}_H{P}_{l+1,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\frac{t^l}{l!}.\end{array}$$

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

$$\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}A(n,l)=\sum _{n=0}^{\mathrm{\infty }}\sum _{l=0}^{n}A(n-l,l),$$

(4.2)

we get

$$\begin{array}{r}\sum _{n=0}^{\mathrm{\infty }}\left[\sum _{l=0}^n{\left(\genfrac{}{}{0ex}{}{n}{l}\right)}_H{P}_{n-l+1,\beta }^{(\alpha )}{(x,y,z;k,a,b)}_H{P}_{l,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\right]\frac{t^n}{n!}\\ =\alpha \sum _{n=0}^{\mathrm{\infty }}\left[\sum _{l=0}^n{\left(\genfrac{}{}{0ex}{}{n}{l}\right)}_H{P}_{n-l,\beta }^{(\alpha )}{(x,y,z;k,a,b)}_H{P}_{l+1,\beta }(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };k,a,b)\right]\frac{t^n}{n!}.\end{array}$$

Whence the result. □

Corollary 4.2 Let $k=a=b=1$ and $\beta =\lambda$. For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following closed form summation formula for the generalized Apostol-Bernoulli polynomials:

$$\begin{array}{r}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)[{}_H\mathcal{B}_{n-k+1}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{B}}_k(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };\lambda )\\ -{\alpha }_H{\mathcal{B}}_{n-k}^{(\alpha )}(x,y,z;\lambda ){\mathcal{B}}_{k+1}(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };\lambda )]=0.\end{array}$$

Corollary 4.3 Let $k+1=-a=b=1$ and $\beta =\lambda$. For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following closed form summation formula for the generalized Apostol-Euler polynomials:

$$\begin{array}{r}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)[{}_H\mathcal{E}_{n-k+1}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{E}}_k(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };\lambda )\\ -{\alpha }_H{\mathcal{E}}_{n-k}^{(\alpha )}(x,y,z;\lambda ){\mathcal{E}}_{k+1}(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };\lambda )]=0.\end{array}$$

Corollary 4.4 Let $k=-2a=b=1$ and $2\beta =\lambda$. For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following closed form summation formula for the generalized Apostol-Genocchi polynomials:

$$\begin{array}{r}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)[{}_H\mathcal{G}_{n-k+1}^{(\alpha )}{(x,y,z;\lambda )}_H{\mathcal{G}}_k(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };\lambda )\\ -{\alpha }_H{\mathcal{G}}_{n-k}^{(\alpha )}(x,y,z;\lambda ){\mathcal{G}}_{k+1}(\frac{x}{\alpha },\frac{y}{\alpha },\frac{z}{\alpha };\lambda )]=0.\end{array}$$

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:

$$\begin{array}{r}{}_{H}P_{n+m,\beta }^{(\alpha )}(X,Y,Z;k,a,b)\\ =\sum _{r,l=0}^{n,m}\left(\genfrac{}{}{0ex}{}{n}{r}\right)\left(\genfrac{}{}{0ex}{}{m}{l}\right)H_{r+l}^{(3)}{(X-x,Y-y,Z-z)}_H{P}_{n+m-r-l}^{(\alpha )}(x,y,z;k,a,b).\end{array}$$

Proof From (2.1), we can write that

$$\begin{array}{rl}{\left(\frac{2^{1-k}{(t+w)}^k}{{\beta }^b{e}^{t+w}-a^b}\right)}^{\alpha }{e}^{x(t+w)+y{(t+w)}^2+z{(t+w)}^3}& ={\sum _{n=0}^{\mathrm{\infty }}}_H{P}_{n,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{{(t+w)}^n}{n!}\\ ={\sum _{n,m=0}^{\mathrm{\infty }}}_H{P}_{n+m,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\frac{w^m}{m!}.\end{array}$$

(4.3)

Therefore, we get

$${\left(\frac{2^{1-k}{(t+w)}^k}{{\beta }^b{e}^{t+w}-a^b}\right)}^{\alpha }=e^{-x(t+w)-y{(t+w)}^2-z{(t+w)}^3}{\sum _{n,m=0}^{\mathrm{\infty }}}_H{P}_{n+m,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\frac{w^m}{m!}.$$

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

$$\begin{array}{r}{\left(\frac{2^{1-k}{(t+w)}^k}{{\beta }^b{e}^{t+w}-a^b}\right)}^{\alpha }{e}^{X(t+w)+Y{(t+w)}^2+Z{(t+w)}^3}\\ =e^{(X-x)(t+w)+(Y-y){(t+w)}^2+(Z-z){(t+w)}^3}{\sum _{n,m=0}^{\mathrm{\infty }}}_H{P}_{n+m,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\frac{w^m}{m!}.\end{array}$$

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

$$\begin{array}{r}{\sum _{n,m=0}^{\mathrm{\infty }}}_H{P}_{n+m,\beta }^{(\alpha )}(X,Y,Z;k,a,b)\frac{t^n}{n!}\frac{w^m}{m!}\\ ={\sum _{n,m=0}^{\mathrm{\infty }}}_H{P}_{n+m,\beta }^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\frac{w^m}{m!}\sum _{r,l=0}^{\mathrm{\infty }}{H}_{r+l}^{(3)}(X-x,Y-y,Z-z)\frac{t^r}{r!}\frac{w^l}{l!}\\ =\sum _{n,m=0}^{\mathrm{\infty }}\sum _{r,l=0}^{n,m}\left(\genfrac{}{}{0ex}{}{n}{r}\right)\left(\genfrac{}{}{0ex}{}{m}{l}\right)H_{r+l}^{(3)}{(X-x,Y-y,Z-z)}_H{P}_{n+m-r-l}^{(\alpha )}(x,y,z;k,a,b)\frac{t^n}{n!}\frac{w^m}{m!}.\end{array}$$

Whence the result. □

Corollary 4.6 Let $k=a=b=1$ and $\beta =\lambda$. For all $c,d,m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following summation formula between the Hermite-based generalized Apostol-Bernoulli polynomials and 3d-Hermite polynomials:

$$\begin{array}{r}{}_H\mathcal{B}_{n+m}^{(\alpha )}(X,Y,Z;\lambda )\\ =\sum _{k,l=0}^{n,m}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{m}{l}\right)H_{k+l}^{(3)}{(X-x,Y-y,Z-z)}_H{\mathcal{B}}_{n+m-k-l}^{(\alpha )}(x,y,z;\lambda ).\end{array}$$

Corollary 4.7 Let $k+1=-a=b=1$ and $\beta =\lambda$. For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following summation formula between the Hermite-based generalized Apostol-Euler polynomials and 3d-Hermite polynomials:

$$\begin{array}{r}{}_H\mathcal{E}_{n+m}^{(\alpha )}(X,Y,Z;\lambda )\\ =\sum _{k,l=0}^{n,m}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{m}{l}\right)H_{k+l}^{(3)}{(X-x,Y-y,Z-z)}_H{\mathcal{E}}_{n+m-k-l}^{(\alpha )}(x,y,z;\lambda ).\end{array}$$

Corollary 4.8 Let $k=-2a=b=1$ and $2\beta =\lambda$. For all $m\in \mathbb{N}$, $n\in {\mathbb{N}}_0$, $\lambda \in \mathbb{C}$, we have the following summation formula between the Hermite-based generalized Apostol-Genocchi polynomials and 3d-Hermite polynomials:

$$\begin{array}{r}{}_H\mathcal{G}_{n+m}^{(\alpha )}(X,Y,Z;\lambda )\\ =\sum _{k,l=0}^{n,m}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{m}{l}\right)H_{k+l}^{(3)}{(X-x,Y-y,Z-z)}_H{\mathcal{G}}_{n+m-k-l}^{(\alpha )}(x,y,z;\lambda ).\end{array}$$