Some properties of the generalized Apostol-type polynomials

Abstract

In this paper, we study some properties of the generalized Apostol-type polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:5702-5728, 2011)), including the recurrence relations, the differential equations and some other connected problems, which extend some known results. We also deduce some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials, and Apostol-Genocchi polynomials of high order.

MSC:11B68, 33C65.

1 Introduction, definitions and motivation

The classical Bernoulli polynomials $B_n(x)$, the classical Euler polynomials $E_n(x)$ and the classical Genocchi polynomials $G_n(x)$, together with their familiar generalizations $B_n^{(\alpha )}(x)$, $E_n^{(\alpha )}(x)$ and $G_n^{(\alpha )}(x)$ of (real or complex) order α, are usually defined by means of the following generating functions (see, for details, [1], pp.532-533 and [2], p.61 et seq.; see also [3] and the references cited therein):

(1.1)

(1.2)

and

$${\left(\frac{2z}{e^z+1}\right)}^{\alpha }\cdot e^{xz}=\sum _{n=0}^{\mathrm{\infty }}{G}_n^{(\alpha )}(x)\frac{z^n}{n!}(|z|<\pi ).$$

(1.3)

So that, obviously, the classical Bernoulli polynomials $B_n(x)$, the classical Euler polynomials $E_n(x)$ and the classical Genocchi polynomials $G_n(x)$ are given, respectively, by

$$B_n(x):=B_n^{(1)}(x),E_n(x):=E_n^{(1)}(x)\text{and}{G}_n(x):=G_n^{(1)}(x)(n\in {\mathbb{N}}_0).$$

(1.4)

For the classical Bernoulli numbers $B_n$, the classical Euler numbers $E_n$ and the classical Genocchi numbers $G_n$ of order n, we have

$$B_n:=B_n(0)=B_n^{(1)}(0),E_n:=E_n(0)=E_n^{(1)}(0)\text{and}{G}_n:=G_n(0)=G_n^{(1)}(0),$$

(1.5)

respectively.

Some interesting analogues of the classical Bernoulli polynomials and numbers were first investigated by Apostol (see [4], p.165, Eq. (3.1)) and (more recently) by Srivastava (see [5], pp.83-84). We begin by recalling here Apostol’s definitions as follows.

Definition 1.1 (Apostol [4]; see also Srivastava [5])

The Apostol-Bernoulli polynomials ${\mathcal{B}}_n(x;\lambda )$ ($\lambda \in \mathbb{C}$) are defined by means of the following generating function:

(1.6)

with, of course,

$$B_n(x)={\mathcal{B}}_n(x;1)\text{and}{\mathcal{B}}_n(\lambda ):={\mathcal{B}}_n(0;\lambda ),$$

(1.7)

where ${\mathcal{B}}_n(\lambda )$ denotes the so-called Apostol-Bernoulli numbers.

Recently, Luo and Srivastava [6] further extended the Apostol-Bernoulli polynomials as the so-called Apostol-Bernoulli polynomials of order α.

Definition 1.2 (Luo and Srivastava [6])

The Apostol-Bernoulli polynomials ${\mathcal{B}}_n^{(\alpha )}(x;\lambda )$ ($\lambda \in \mathbb{C}$) of order $\alpha \in {\mathbb{N}}_0$ are defined by means of the following generating function:

(1.8)

with, of course,

$$B_n^{(\alpha )}(x)={\mathcal{B}}_n^{(\alpha )}(x;1)\text{and}{\mathcal{B}}_n^{(\alpha )}(\lambda ):={\mathcal{B}}_n^{(\alpha )}(0;\lambda ),$$

(1.9)

where ${\mathcal{B}}_n^{(\alpha )}(\lambda )$ denotes the so-called Apostol-Bernoulli numbers of order α.

On the other hand, Luo [7], gave an analogous extension of the generalized Euler polynomials as the so-called Apostol-Euler polynomials of order α.

Definition 1.3 (Luo [7])

The Apostol-Euler polynomials ${\mathcal{E}}_n^{(\alpha )}(x;\lambda )$ ($\lambda \in \mathbb{C}$) of order $\alpha \in {\mathbb{N}}_0$ are defined by means of the following generating function:

$${\left(\frac{2}{\lambda e^z+1}\right)}^{\alpha }\cdot e^{xz}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_n^{(\alpha )}(x;\lambda )\frac{z^n}{n!}(|z|<|log(-\lambda )|)$$

(1.10)

with, of course,

$$E_n^{(\alpha )}(x)={\mathcal{E}}_n^{(\alpha )}(x;1)\text{and}{\mathcal{E}}_n^{(\alpha )}(\lambda ):={\mathcal{E}}_n^{(\alpha )}(0;\lambda ),$$

(1.11)

where ${\mathcal{E}}_n^{(\alpha )}(\lambda )$ denotes the so-called Apostol-Euler numbers of order α.

On the subject of the Genocchi polynomials $G_n(x)$ and their various extensions, a remarkably large number of investigations have appeared in the literature (see, for example, [8–14]). Moreover, Luo (see [12–14]) introduced and investigated the Apostol-Genocchi polynomials of (real or complex) order α, which are defined as follows:

Definition 1.4 The Apostol-Genocchi polynomials ${\mathcal{G}}_n^{(\alpha )}(x;\lambda )$ ($\lambda \in \mathbb{C}$) of order $\alpha \in {\mathbb{N}}_0$ are defined by means of the following generating function:

$${\left(\frac{2z}{\lambda e^z+1}\right)}^{\alpha }\cdot e^{xz}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_n^{(\alpha )}(x;\lambda )\frac{z^n}{n!}(|z|<|log(-\lambda )|)$$

(1.12)

with, of course,

$$\begin{array}{r}{G}_n^{(\alpha )}(x)={\mathcal{G}}_n^{(\alpha )}(x;1),{\mathcal{G}}_n^{(\alpha )}(\lambda ):={\mathcal{G}}_n^{(\alpha )}(0;\lambda ),\\ {\mathcal{G}}_n(x;\lambda ):={\mathcal{G}}_n^{(1)}(x;\lambda )\text{and}{\mathcal{G}}_n(\lambda ):={\mathcal{G}}_n^{(1)}(\lambda ),\end{array}$$

(1.13)

where ${\mathcal{G}}_n(\lambda )$, ${\mathcal{G}}_n^{(\alpha )}(\lambda )$ and ${\mathcal{G}}_n(x;\lambda )$ denote the so-called Apostol-Genocchi numbers, the Apostol-Genocchi numbers of order α and the Apostol-Genocchi polynomials, respectively.

Recently, Luo and Srivastava [15] introduced a unification (and generalization) of the above-mentioned three families of the generalized Apostol type polynomials.

Definition 1.5 (Luo and Srivastava [15])

The generalized Apostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$ ($\alpha \in {\mathbb{N}}_0$, $\lambda ,u,v\in \mathbb{C}$) of order α are defined by means of the following generating function:

$${\left(\frac{2^u{z}^v}{\lambda e^z+1}\right)}^{\alpha }\cdot e^{xz}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)\frac{z^n}{n!}(|z|<|log(-\lambda )|),$$

(1.14)

where

$${\mathcal{F}}_n^{(\alpha )}(\lambda ;u,v):={\mathcal{F}}_n^{(\alpha )}(0;\lambda ;u,v)$$

(1.15)

denote the so-called Apostol type numbers of order α.

So that, by comparing Definition 1.5 with Definitions 1.2, 1.3 and 1.4, we have

(1.16)

(1.17)

(1.18)

A polynomial $p_n(x)$ ($n\in \mathbb{N}$, $x\in \mathbb{C}$) is said to be a quasi-monomial [16], whenever two operators $\hat{M}$, $\hat{P}$, called multiplicative and derivative (or lowering) operators respectively, can be defined in such a way that

(1.19)

(1.20)

which can be combined to get the identity

$$\hat{M}\hat{P}{p}_n(x)=n{p}_n(x).$$

(1.21)

The Appell polynomials [17] can be defined by considering the following generating function:

$$A(t)e^{xt}=\sum _{n=0}^{\mathrm{\infty }}\frac{R_n(x)}{n!}{t}^n,$$

(1.22)

where

$$A(t)=\sum _{k=0}^{\mathrm{\infty }}\frac{R_k}{k!}{t}^k(A(0)\ne 0)$$

(1.23)

is analytic function at $t=0$.

From [18], we know that the multiplicative and derivative operators of $R_n(x)$ are

(1.24)

(1.25)

where

$$\frac{A^{\prime }(t)}{A(t)}=\sum _{n=0}^{\mathrm{\infty }}{\alpha }_n\frac{t^n}{n!}.$$

(1.26)

By using (1.21), we have the following lemma.

Lemma 1.6 ([18])

The Appell polynomials $R_n(x)$ defined by (1.22) satisfy the differential equation:

$$\frac{{\alpha }_{n-1}}{(n-1)!}{y}^{(n)}+\frac{{\alpha }_{n-2}}{(n-2)!}{y}^{(n-1)}+\cdots +\frac{{\alpha }_1}{1!}{y}^{″}+(x+{\alpha }_0)y^{\prime }-ny=0,$$

(1.27)

where the numerical coefficients ${\alpha }_k$, $k=1,2,\dots ,n-1$ are defined in (1.26), and are linked to the values $R_k$ by the following relations:

$$R_{k+1}=\sum _{h=0}^k\left(\begin{array}{c}k\\ h\end{array}\right)R_h{\alpha }_{k-h}.$$

Let $\mathcal{P}$ be the vector space of polynomials with coefficients in ℂ. A polynomial sequence ${\{P_n\}}_{n\ge 0}$ be a polynomial set. ${\{P_n\}}_{n\ge 0}$ is called a σ-Appell polynomial set of transfer power series A is generated by

$$G(x,t)=A(t)G_0(x,t)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n(x)}{n!}{t}^n,$$

(1.28)

where $G_0(x,t)$ is a solution of the system:

In [19], the authors investigated the connection coefficients between two polynomials. And there is a result about connection coefficients between two σ-Appell polynomial sets.

Lemma 1.7 ([19])

Let $\sigma \in {\mathrm{\Lambda }}^{(-1)}$. Let ${\{P_n\}}_{n\ge 0}$ and ${\{Q_n\}}_{n\ge 0}$ be two σ-Appell polynomial sets of transfer power series, respectively, $A_1$ and $A_2$. Then

$$Q_n(x)=\sum _{m=0}^n\frac{n!}{m!}{\alpha }_{n-m}{P}_m(x),$$

(1.29)

where

$$\frac{A_2(t)}{A_1(t)}=\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k{t}^k.$$

In recent years, several authors obtained many interesting results involving the related Bernoulli polynomials and Euler polynomials [5, 20–40]. And in [29], the authors studied some series identities involving the generalized Apostol type and related polynomials.

In this paper, we study some other properties of the generalized Apostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$, including the recurrence relations, the differential equations and some connection problems, which extend some known results. As special, we obtain some properties of the generalized Apostol-Euler polynomials, the generalized Apostol-Bernoulli polynomials and Apostol-Genocchi polynomials of high order.

2 Recursion formulas and differential equations

From the generating function (1.14), we have

$$\frac{\partial }{\partial x}{\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=n{\mathcal{F}}_{n-1}^{(\alpha )}(x;\lambda ;u,v).$$

(2.1)

A recurrence relation for the generalized Apostol type polynomials is given by the following theorem.

Theorem 2.1 For any integral $n\ge 1$, $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{N}$, the following recurrence relation for the generalized Apostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$ holds true:

$$(\frac{\alpha v}{n+1}-1){\mathcal{F}}_{n+1}^{(\alpha )}(x;\lambda ;u,v)=\frac{\alpha \lambda }{2^u}\cdot \frac{n!}{(n+v)!}{\mathcal{F}}_{n+v}^{(\alpha +1)}(x+1;\lambda ;u,v)-x{\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v).$$

(2.2)

Proof Differentiating both sides of (1.14) with respect to t, and using some elementary algebra and the identity principle of power series, recursion (2.2) easily follows. □

By setting $\lambda :=-\lambda$, $u=0$ and $v=1$ in Theorem 2.1, and then multiplying ${(-1)}^{\alpha }$ on both sides of the result, we have:

Corollary 2.2 For any integral $n\ge 1$, $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{N}$, the following recurrence relation for the generalized Apostol-Bernoulli polynomials ${\mathcal{B}}_n^{(\alpha )}(x;\lambda )$ holds true:

$$[\alpha -(n+1)]{\mathcal{B}}_{n+1}^{(\alpha )}(x;\lambda )=\alpha \lambda {\mathcal{B}}_{n+1}^{(\alpha +1)}(x+1;\lambda )-x{\mathcal{B}}_n^{(\alpha )}(x;\lambda ).$$

(2.3)

By setting $u=1$ and $v=0$ in Theorem 2.1, we have the following corollary.

Corollary 2.3 For any integral $n\ge 1$, $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{N}$, the following recurrence relation for the generalized Apostol-Euler polynomials ${\mathcal{E}}_n^{(\alpha )}(x;\lambda )$ holds true:

$${\mathcal{E}}_{n+1}^{(\alpha )}(x;\lambda )=x{\mathcal{E}}_n^{(\alpha )}(x;\lambda )-\frac{\alpha \lambda }{2}{\mathcal{E}}_n^{(\alpha +1)}(x+1;\lambda ).$$

(2.4)

By setting $u=1$ and $v=1$ in Theorem 2.1, we have the following corollary.

Corollary 2.4 For any integral $n\ge 1$, $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{N}$, the following recurrence relation for the generalized Apostol-Genocchi polynomials ${\mathcal{G}}_n^{(\alpha )}(x;\lambda )$ holds true:

$$2[\alpha -(n+1)]{\mathcal{G}}_{n+1}^{(\alpha )}(x;\lambda )=\alpha \lambda {\mathcal{G}}_{n+1}^{(\alpha +1)}(x+1;\lambda )-2(n+1)x{\mathcal{G}}_n^{(\alpha )}(x;\lambda ).$$

(2.5)

From (1.14) and (1.22), we know that the generalized Appostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$ is Appell polynomials with

$$A(t)={\left(\frac{2^u{t}^v}{\lambda e^t+1}\right)}^{\alpha }.$$

(2.6)

From the Eq. (23) of [15], we know that ${\mathcal{G}}_0(1;\lambda )=0$. So from (2.6) and (1.12), we can obtain that if $v=0$, we have

$$\frac{A^{\prime }(t)}{A(t)}=\frac{\lambda \alpha }{2}\sum _{n=0}^{\mathrm{\infty }}\frac{{\mathcal{G}}_{n+1}(1;\lambda )}{n+1}\cdot \frac{t^n}{n!}.$$

(2.7)

By using (1.24) and (1.26), we can obtain the multiplicative and derivative operators of the generalized Appostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$

(2.8)

(2.9)

From (2.1), we can obtain

$$\frac{{\partial }^p}{\partial x^p}{\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=\frac{n!}{(n-p)!}{\mathcal{F}}_{n-p}^{(\alpha )}(x;\lambda ;u,v).$$

(2.10)

Then by using (1.20), (2.8) and (2.10), we obtain the following result.

Theorem 2.5 For any integral $n\ge 1$, $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{N}$, the following recurrence relation for the generalized Apostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,0)$ holds true:

$$\begin{array}{rcl}{\mathcal{F}}_{n+1}^{(\alpha )}(x;\lambda ;u,0)& =& (x+\frac{\lambda \alpha }{2}{\mathcal{G}}_1(1;\lambda )){\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,0)\\ +\frac{\lambda \alpha }{2}\sum _{k=0}^{n-1}\left(\begin{array}{c}n\\ k\end{array}\right)\frac{{\mathcal{G}}_{n-k+1}(1;\lambda )}{n-k+1}{\mathcal{F}}_{n-k}^{(\alpha )}(x;\lambda ;u,0).\end{array}$$

(2.11)

By setting $u=1$ in Theorem 2.5, we have the following corollary.

Corollary 2.6 For any integral $n\ge 1$, $\lambda \in \mathbb{C}$ and $\alpha \in \mathbb{N}$, the following recurrence relation for the generalized Apostol-Euler polynomials ${\mathcal{E}}_n^{(\alpha )}(x;\lambda )$ holds true:

$${\mathcal{E}}_{n+1}^{(\alpha )}(x;\lambda )=(x+\frac{\lambda \alpha }{2}{\mathcal{G}}_1(1;\lambda )){\mathcal{E}}_n^{(\alpha )}(x;\lambda )+\frac{\lambda \alpha }{2}\sum _{k=0}^{n-1}\left(\begin{array}{c}n\\ k\end{array}\right)\frac{{\mathcal{G}}_{n-k+1}(1;\lambda )}{n-k+1}{\mathcal{E}}_{n-k}^{(\alpha )}(x;\lambda ).$$

(2.12)

Furthermore, applying Lemma 1.7 to ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,0)$, we have the following theorem.

Theorem 2.7 The generalized Apostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,0)$ satisfy the differential equation:

(2.13)

Specially, by setting $u=1$ in Theorem 2.7, then we have the following corollary.

Corollary 2.8 The generalized Apostol-Euler polynomials ${\mathcal{E}}_n^{(\alpha )}(x;\lambda )$ satisfy the differential equation:

(2.14)

3 Connection problems

From (1.14) and (1.28), we know that the generalized Apostol type polynomials ${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$ are a $D_x$-Appell polynomial set, where $D_x$ denotes the derivative operator.

From Table 1 in [19], we know that the derivative operators of monomials $x^n$ and the Gould-Hopper polynomials $g_n^m(x,h)$ [30] are all $D_x$. And their transfer power series $A(t)$ are 1 and $e^{h{t}^m}$, respectively.

Applying Lemma 1.7 to $P_n(x)=x^n$ and $Q_n(x)={\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

Theorem 3.1

$${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{F}}_{n-m}^{(\alpha )}(\lambda ;u,v)x^m,$$

(3.1)

where ${\mathcal{F}}_n^{(\alpha )}(\lambda ;u,v)$ is the so-called Apostol type numbers of order α defined by (1.15).

By setting $\lambda :=-\lambda$, $u=0$ and $v=1$ in Theorem 3.1, and then multiplying ${(-1)}^{\alpha }$ on both sides of the result, we have the following corollary.

Corollary 3.2

$${\mathcal{B}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{B}}_{n-m}^{(\alpha )}(\lambda )x^m,$$

(3.2)

which is just Eq. (3.1) of [23].

By setting $u=0$ and $v=0$ in Theorem 3.1, we have the following corollary.

Corollary 3.3

$${\mathcal{E}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{E}}_{n-m}^{(\alpha )}(\lambda )x^m.$$

(3.3)

By setting $u=1$ and $v=1$ in Theorem 3.1, we have the following corollary.

Corollary 3.4

$${\mathcal{G}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{G}}_{n-m}^{(\alpha )}(\lambda )x^m,$$

(3.4)

which is just Eq. (24) of [15].

Applying Lemma 1.7 to $P_n(x)={\mathcal{F}}_n(x;\lambda ;u,v)$ and $Q_n(x)={\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

Theorem 3.5

$${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{F}}_{n-m}^{(\alpha -1)}(\lambda ;u,v){\mathcal{F}}_m(x;\lambda ;u,v),$$

(3.5)

where ${\mathcal{F}}_n^{(\alpha )}(\lambda ;u,v)$ is the so-called Apostol type numbers of order α defined by (1.15).

By setting $\lambda :=-\lambda$, $u=0$ and $v=1$ in Theorem 3.5, and then multiplying ${(-1)}^{\alpha }$ on both sides of the result, we have the following corollary.

Corollary 3.6

$${\mathcal{B}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{B}}_{n-m}^{(\alpha -1)}(\lambda ){\mathcal{B}}_m(x;\lambda ),$$

(3.6)

which is just Eq. (3.2) of [23].

By setting $u=1$ and $v=0$ in Theorem 3.5, we have the following corollary.

Corollary 3.7

$${\mathcal{E}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{E}}_{n-m}^{(\alpha -1)}(\lambda ){\mathcal{E}}_m(x;\lambda ).$$

(3.7)

By setting $u=1$ and $v=1$ in Theorem 3.5, we have the following corollary.

Corollary 3.8

$${\mathcal{G}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{G}}_{n-m}^{(\alpha -1)}(\lambda ){\mathcal{G}}_m(x;\lambda ).$$

(3.8)

Applying Lemma 1.7 to $P_n(x)=g_n^m(x,h)$ and $Q_n(x)={\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

Theorem 3.9

$${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=\sum _{r=0}^n\frac{n!}{r!}[\sum _{k=0}^{[(n-r)/m]}{(-1)}^k\frac{h^k}{k!(n-r-mk)!}{\mathcal{F}}_{n-r-mk}^{(\alpha )}(\lambda ;u,v)]g_r^m(x,h).$$

(3.9)

By setting $\lambda :=-\lambda$, $u=0$ and $v=1$ in Theorem 3.9, and then multiplying ${(-1)}^{\alpha }$ on both sides of the result, we have the following corollary.

Corollary 3.10

$${\mathcal{B}}_n^{(\alpha )}(x;\lambda )=\sum _{r=0}^n\frac{n!}{r!}[\sum _{k=0}^{[(n-r)/m]}{(-1)}^k\frac{h^k}{k!(n-r-mk)!}{\mathcal{B}}_{n-r-mk}^{(\alpha )}(\lambda )]g_r^m(x,h),$$

(3.10)

which is just Eq. (3.3) of [23].

By setting $u=1$ and $v=0$ in Theorem 3.9, we have the following corollary.

Corollary 3.11

$${\mathcal{E}}_n^{(\alpha )}(x;\lambda )=\sum _{r=0}^n\frac{n!}{r!}[\sum _{k=0}^{[(n-r)/m]}{(-1)}^k\frac{h^k}{k!(n-r-mk)!}{\mathcal{E}}_{n-r-mk}^{(\alpha )}(\lambda )]g_r^m(x,h).$$

(3.11)

By setting $u=1$ and $v=1$ in Theorem 3.9, we have the following corollary.

Corollary 3.12

$${\mathcal{G}}_n^{(\alpha )}(x;\lambda )=\sum _{r=0}^n\frac{n!}{r!}[\sum _{k=0}^{[(n-r)/m]}{(-1)}^k\frac{h^k}{k!(n-r-mk)!}{\mathcal{G}}_{n-r-mk}^{(\alpha )}(\lambda )]g_r^m(x,h).$$

(3.12)

When $v\alpha =1$, applying Lemma 1.7 to $P_n(x)={\mathcal{E}}_n^{(\alpha -1)}(x;\lambda )$ and $Q_n(x)={\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$, we have the following theorem.

Theorem 3.13 If $v\alpha =1$, then we have

$${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right)2^{(u-1)\alpha }{\mathcal{G}}_{n-m}(\lambda ){\mathcal{E}}_m^{(\alpha -1)}(x;\lambda ).$$

(3.13)

By setting $\lambda :=-\lambda$, $u=0$ and $v=1$ in Theorem 3.13, and then multiplying ${(-1)}^{\alpha }$ on both sides of the result, we have the following corollary.

Corollary 3.14

$${\mathcal{B}}_n(x;\lambda )=-\frac{1}{2}\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{G}}_{n-m}(-\lambda )x^m.$$

(3.14)

By setting $u=1$ and $v=1$ in Theorem 3.13, we have the following corollary.

Corollary 3.15

$${\mathcal{G}}_n(x;\lambda )=-\frac{1}{2}\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{G}}_{n-m}(\lambda )x^m,$$

(3.15)

which is just the case of $\alpha =1$ in (3.4).

When $v=1$ or $\alpha =0$, applying Lemma 1.7 to $P_n(x)={\mathcal{G}}_n^{(\alpha -1)}(x;\lambda )$ and $Q_n(x)={\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)$, we can obtain the following theorem.

Theorem 3.16 If $v=1$ or $\alpha =0$, we have

$${\mathcal{F}}_n^{(\alpha )}(x;\lambda ;u,v)=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right)2^{(u-1)\alpha }{\mathcal{G}}_{n-m}(\lambda ){\mathcal{G}}_m^{(\alpha -1)}(x;\lambda ).$$

(3.16)

By setting $\lambda :=-\lambda$, $u=0$ and $v=1$ in Theorem 3.13, and then multiplying ${(-1)}^{\alpha }$ on both sides of the result, we have the following corollary.

Corollary 3.17

$${\mathcal{B}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){(-\frac{1}{2})}^{\alpha }{\mathcal{G}}_{n-m}(-\lambda ){\mathcal{G}}_m^{(\alpha -1)}(x;-\lambda ).$$

(3.17)

When $\alpha =1$ in (3.17), it is just (3.15).

By setting $u=1$ and $v=1$ in Theorem 3.16, we have the following corollary.

Corollary 3.18

$${\mathcal{G}}_n^{(\alpha )}(x;\lambda )=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{G}}_{n-m}(\lambda ){\mathcal{G}}_m^{(\alpha -1)}(x;\lambda ),$$

(3.18)

which is equal to (3.8).

If $\alpha =0$ in Theorem 3.16, we have:

Corollary 3.19

$$x^n=\sum _{m=0}^n\left(\begin{array}{c}n\\ m\end{array}\right){\mathcal{G}}_{n-m}(\lambda ){\mathcal{G}}_m^{(-1)}(x;\lambda ).$$

(3.19)

4 Hermite-based generalized Apostol type polynomials

Finally, we give a generation of the generalized Apostol type polynomials.

The two-variable Hermite-Kampé de Fériet polynomials (2VHKdFP) $H_n(x,y)$ are defined by the series [31]

$$H_n(x,y)=n!\sum _{r=0}^{[n/2]}\frac{x^{n-2r}{y}^r}{r!(n-2r)!}$$

(4.1)

with the following generating function:

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

(4.2)

And the 2VHKdFP $H_n(x,y)$ are also defined through the operational identity

$$exp\left(y\frac{{\partial }^2}{\partial x^2}\right)\left\{x^n\right\}=H_n(x,y).$$

(4.3)

Acting the operator $exp(y\frac{{\partial }^2}{\partial x^2})$ on (1.14), and by the identity [32]

$$exp\left(y\frac{{\partial }^2}{\partial x^2}\right)\{exp(-a{x}^2+bx)\}=\frac{1}{\sqrt{1+4ay}}exp(-\frac{a{x}^2-bx-b^{2}y}{1+4ay}),$$

(4.4)

we define the Hermite-based generalized Apostol type polynomials ${}_H\mathcal{F}_n^{(\alpha )}(x,y;\lambda ;u,v)$ by the generating function

$${\left(\frac{2^u{z}^v}{\lambda e^t+1}\right)}^{\alpha }\cdot e^{xt+y{t}^2}={\sum _{n=0}^{\mathrm{\infty }}}_H{\mathcal{F}}_n^{(\alpha )}(x,y;\lambda ;u,v)\frac{t^n}{n!}(|t|<|log(-\lambda )|).$$

(4.5)

Clearly, we have

$${}_H\mathcal{F}_n(x,y;\lambda ;u,v){=}_H{\mathcal{F}}_n^{(1)}(x,y;\lambda ;u,v).$$

From the generating function (4.5), we easily obtain

$${\frac{\partial }{\partial x}}_H{\mathcal{F}}_n^{(\alpha )}(x,y;\lambda ;u,v)=n_H{\mathcal{F}}_{n-1}^{(\alpha )}(x,y;\lambda ;u,v)$$

(4.6)

and

$${\frac{\partial }{\partial y}}_H{\mathcal{F}}_n^{(\alpha )}(x,y;\lambda ;u,v)=n{(n-1)}_H{\mathcal{F}}_{n-2}^{(\alpha )}(x,y;\lambda ;u,v),$$

(4.7)

which can be combined to get the identity

$${\frac{{\partial }^2}{\partial x^2}}_H{\mathcal{F}}_n^{(\alpha )}(x,y;\lambda ;u,v)={\frac{\partial }{\partial y}}_H{\mathcal{F}}_n^{(\alpha )}(x,y;\lambda ;u,v).$$

(4.8)

Acting with the operator $expy\frac{{\partial }^2}{\partial x^2}$ on both sides of (3.1), (3.5), (3.13), (3.18), and by using (4.3), we obtain

(4.9)

(4.10)

(4.11)

(4.12)

where ${}_H\mathcal{E}_n^{(\alpha )}(x;\lambda )$ and ${}_H\mathcal{G}_n^{(\alpha )}(x;\lambda )$ are the Hermite-based generalized Apostol-Euler polynomials and the Hermite-based generalized Apostol-Genocchi polynomials respectively, defined by the following generating functions: