1 Introduction, definitions, and motivation

Throughout this paper, we always make use of the following notations: \(\mathbb{N}=\{1,2,3, \ldots\}\) denotes the set of natural numbers, \(\mathbb{N}_{0}=\{0,1,2,3, \ldots\}\) denotes the set of nonnegative integers, \(\mathbb{Z}_{0}^{-}=\{0,-1,-2,-3,\ldots\}\) denotes the set of nonpositive integers, \(\mathbb{Z}\) denotes the set of integers, \(\mathbb{R}\) denotes the set of real numbers, and \(\mathbb{C}\) denotes the set of complex numbers.

The symbol \((a)_{k}\) denotes the shifted factorial (or the Pochhammer symbol), defined, \(a \in\mathbb{C}\), by

$$ (a)_{k} = \frac{\Gamma(a+k)}{\Gamma(a)} = \begin{cases} 1, & k=0,\\ a(a+1) \cdots(a+k-1), & k \in\mathbb{N}. \end{cases} $$
(1.1)

The symbol \(\{n\}_{k}\) denotes the falling factorial, defined, \(a \in \mathbb{C}\), by

$$ \{a\}_{k}= \begin{cases} 1, & k=0, \\ a(a-1)\cdots(a-k+1)=\frac{\Gamma(a+1)}{\Gamma(a-k+1)}, & k \in \mathbb{N}, \end{cases} $$
(1.2)

where \(\Gamma(x)\) is the usual gamma function.

The classical Bernoulli polynomials \(B_{n}(x)\), Euler polynomials \(E_{n}(x)\), and Genocchi polynomials \(G_{n}(x)\), together with their familiar generalizations \(B_{n}^{(\alpha)}(x)\), \(E_{n}^{(\alpha)}(x)\), and \(G_{n}^{(\alpha)}(x)\) of order α, are usually defined by means of the following generating functions (see, for details, [1], pp.532-533 and [2]):

$$\begin{aligned}& \biggl(\frac{z}{e^{z}-1} \biggr)^{\alpha}e^{xz}=\sum _{n=0}^{\infty }B_{n}^{(\alpha)}(x) \frac{z^{n}}{n!}\quad \bigl(\vert z\vert < 2\pi\bigr), \end{aligned}$$
(1.3)
$$\begin{aligned}& \biggl(\frac{2}{e^{z}+1} \biggr)^{\alpha}e^{xz}=\sum _{n=0}^{\infty }E_{n}^{(\alpha)}(x) \frac{z^{n}}{n!} \quad \bigl(\vert z\vert < \pi\bigr) \end{aligned}$$
(1.4)

and

$$ \biggl(\frac{2z}{e^{z}+1} \biggr)^{\alpha}e^{xz}=\sum _{n=0}^{\infty }G_{n}^{(\alpha)}(x) \frac{z^{n}}{n!} \quad \bigl(\vert z\vert < \pi\bigr). $$
(1.5)

Thus, the Bernoulli polynomials \(B_{n}(x)\), Euler polynomials \(E_{n}(x)\), and Genocchi polynomials \(G_{n}(x)\) are given, respectively, by

$$ B_{n}(x):=B_{n}^{(1)}(x), \qquad E_{n}(x):=E_{n}^{(1)}(x) \quad \text{and}\quad G_{n}(x):=G_{n}^{(1)}(x) \quad (n\in{ \mathbb{N}}_{0}). $$
(1.6)

The Bernoulli numbers \(B_{n}\), Euler numbers \(E_{n}\), and Genocchi numbers \(G_{n}\) are, respectively,

$$ B_{n}:=B_{n}(0)=B_{n}^{(1)}(0), \qquad E_{n}:=E_{n}(0)=E_{n}^{(1)}(0) \quad \text {and}\quad G_{n}:=G_{n}(0)=G_{n}^{(1)}(0). $$
(1.7)

Some interesting analogs of the classical Bernoulli polynomials and numbers were first investigated by Apostol (see [3], p.165, Eq. (3.1)) and (more recently) by Srivastava (see [4], pp.83-84). We begin by recalling here Apostol’s definitions as follows.

Definition 1.1

(Apostol [3]; see also Srivastava [4])

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

$$ \frac{ze^{xz}}{\lambda e^{z}-1}=\sum_{n=0}^{\infty}{ \mathcal {B}}_{n}(x;\lambda)\frac{z^{n}}{n!} \quad \bigl(\vert z\vert < 2\pi \mbox{ when } \lambda=1; |z|<|\log{\lambda}| \mbox{ when } \lambda \neq1\bigr) $$
(1.8)

with, of course,

$$ B_{n}(x)={\mathcal{B}}_{n}(x;1)\quad \mbox{and} \quad { \mathcal {B}}_{n}(\lambda):={\mathcal{B}}_{n}(0;\lambda), $$
(1.9)

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

Recently, Luo and Srivastava [5] further extended the Apostol-Bernoulli polynomials as the so-called Apostol-Bernoulli polynomials of order α.

Definition 1.2

(Luo and Srivastava [5])

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

$$\begin{aligned}& { \biggl(\frac{z}{\lambda e^{z}-1} \biggr)}^{\alpha}\cdot e^{xz}=\sum _{n=0}^{\infty}{\mathcal {B}}_{n}^{(\alpha)}(x; \lambda)\frac{z^{n}}{n!} \\& \quad \bigl(\vert z\vert < 2\pi \mbox{ when } \lambda=1; |z|<|\log{\lambda}| \mbox{ when } \lambda \neq1\bigr) \end{aligned}$$
(1.10)

with, of course,

$$ B_{n}^{(\alpha)}(x)={\mathcal {B}}_{n}^{(\alpha)}(x;1) \quad \mbox{and} \quad {\mathcal {B}}_{n}^{(\alpha)}(\lambda):={ \mathcal{B}}_{n}^{(\alpha)}(0;\lambda), $$
(1.11)

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

On the other hand, Luo [6] gave an analogous extension of the generalized Euler polynomials as the so-called Apostol-Euler polynomials of order α.

Definition 1.3

(Luo [6])

The Apostol-Euler polynomials \({\mathcal{E}}_{n}^{(\alpha)}(x;\lambda)\) of order α (\(\alpha, \lambda\in\mathbb{C}\)) are defined by means of the following generating function:

$$ { \biggl(\frac{2}{\lambda e^{z}+1} \biggr)}^{\alpha}\cdot e^{xz}=\sum _{n=0}^{\infty}{\mathcal {E}}_{n}^{(\alpha)}(x; \lambda)\frac{z^{n}}{n!} \quad \bigl(\vert z\vert < \bigl\vert \log ({- \lambda})\bigr\vert \bigr) $$
(1.12)

with, of course,

$$ E_{n}^{(\alpha)}(x)={\mathcal {E}}_{n}^{(\alpha)}(x;1) \quad \mbox{and}\quad {\mathcal {E}}_{n}^{(\alpha)}(\lambda):={ \mathcal{E}}_{n}^{(\alpha)}(0;\lambda), $$
(1.13)

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, [7–11]). Moreover, Luo (see [12]) 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 α (\(\alpha\in\mathbb{N}\)) are defined by means of the following generating function:

$$ { \biggl(\frac{2z}{\lambda e^{z}+1} \biggr)}^{\alpha}\cdot e^{xz}=\sum _{n=0}^{\infty}{\mathcal {G}}_{n}^{(\alpha)}(x; \lambda)\frac{z^{n}}{n!} \quad \bigl(|z|< \bigl\vert \log ({-\lambda})\bigr\vert \bigr) $$
(1.14)

with, of course,

$$ \begin{aligned} &G_{n}^{(\alpha)}(x)={\mathcal{G}}_{n}^{(\alpha)}(x;1), \qquad {\mathcal {G}}_{n}^{(\alpha)}(\lambda):={\mathcal {G}}_{n}^{(\alpha)}(0;\lambda), \\ &{\mathcal {G}}_{n}(x;\lambda):={\mathcal {G}}_{n}^{(1)}(x; \lambda) \quad \mbox{and} \quad {\mathcal {G}}_{n}(\lambda):={ \mathcal{G}}_{n}^{(1)}(\lambda), \end{aligned} $$
(1.15)

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.

Ozden et al. [13] investigated the following unification (and generalization) of the generating functions of the three families of Apostol-type polynomials:

$$\begin{aligned}& \frac{2^{1-\kappa}z^{\kappa}}{{\beta}^{b}e^{z}-a^{b}}e^{xz}=\sum _{n=0}^{\infty}{\mathcal {Y}}_{n,\beta}(x; \kappa,a,b)\frac{z^{n}}{n!} \\& \quad \bigl(|z|< 2\pi \mbox{ when } \beta=a; |z|<\bigl\vert b\log(\beta /a) \bigr\vert \mbox{ when } \beta\neq a; \kappa,\beta\in\mathbb{C}; a,b \in \mathbb{C}\setminus\{0\} \bigr). \end{aligned}$$
(1.16)

In [14] Özarslan further gave an extension of the above definition (1.16) as follows:

$$\begin{aligned}& \biggl(\frac{2^{1-\kappa}z^{\kappa}}{{\beta}^{b}e^{z}-a^{b}} \biggr)^{\alpha}e^{xz}= \sum_{n=0}^{\infty}{\mathcal {Y}}^{(\alpha)}_{n,\beta}(x;\kappa,a,b)\frac{z^{n}}{n!} \\& \quad \bigl(\alpha\in\mathbb{N}; |z|< 2\pi \mbox{ when } \beta=a; |z|<\bigl\vert b\log(\beta /a)\bigr\vert \\& \quad \mbox{when } \beta\neq a; \kappa,\beta\in \mathbb{C}; a,b\in \mathbb{C}\setminus\{0\} \bigr) \end{aligned}$$
(1.17)

and gave some identities for \({\mathcal{Y}}^{(\alpha)}_{n,\beta }(x;\kappa,a,b)\).

Recently, Luo and Srivastava [15] further extended the Apostol-type polynomials as follows.

Definition 1.5

(Luo and Srivastava [15])

The generalized Apostol-type polynomials \({\mathcal{F}}_{n}^{(\alpha )}(x;\lambda;\mu;\nu)\) of order α (\(\alpha,\lambda,\mu ;\nu\in\mathbb{C}\)) are defined by means of the following generating function:

$$ { \biggl(\frac{2^{\mu}z^{\nu}}{\lambda e^{z}+1} \biggr)}^{\alpha} e^{xz}=\sum_{n=0}^{\infty}{\mathcal {F}}_{n}^{(\alpha)}(x;\lambda;\mu;\nu)\frac{z^{n}}{n!}\quad \bigl(|z|< \bigl\vert \log{(-\lambda)}\bigr\vert \bigr). $$
(1.18)

By comparing Definition 1.5 with Definitions 1.2, 1.3 and 1.4, we readily find that

$$\begin{aligned}& \mathcal{B}_{n}^{ ( \alpha ) } ( x;\lambda )=(-1)^{\alpha}\mathcal{F}_{n}^{ ( \alpha ) } (x;-\lambda; 0;1 ) \quad (\alpha\in\mathbb{N}), \end{aligned}$$
(1.19)
$$\begin{aligned}& \mathcal{E}_{n}^{ ( \alpha ) } ( x;\lambda )= \mathcal{F}_{n}^{ ( \alpha ) } (x;\lambda; 1;0 ) \quad (\alpha\in \mathbb{C}) \end{aligned}$$
(1.20)

and

$$ \mathcal{G}_{n}^{ ( \alpha ) } ( x;\lambda )= \mathcal{F}_{n}^{ ( \alpha ) } ( x;\lambda; 1;1 ) \quad (\alpha\in \mathbb{N}) . $$
(1.21)

Furthermore, if we compare the generating functions (1.16), (1.17) and (1.18), we readily see that

$$\begin{aligned}& {\mathcal{Y}}_{n,\beta}(x;\kappa,a,b)=-\frac{1}{a^{b}}{ \mathcal {F}}_{n}^{(1)} \biggl(x;- \biggl(\frac{\beta}{a} \biggr)^{b};1-\kappa ;\kappa \biggr), \end{aligned}$$
(1.22)
$$\begin{aligned}& {\mathcal{Y}}^{(\alpha)}_{n,\beta}(x; \kappa,a,b)=(-1)^{\alpha }\frac{1}{a^{b \alpha}}{\mathcal {F}}_{n}^{(\alpha)} \biggl(x;- \biggl(\frac{\beta}{a} \biggr)^{b};1-\kappa ;\kappa \biggr). \end{aligned}$$
(1.23)

More investigations of this subject can be found in [5, 6, 12–22].

The aim of this paper is to give the multiplication formula for the Apostol-type polynomials \(\mathcal{F}_{n}^{ ( \alpha ) } (x;\lambda; \mu;\nu )\) and obtain an explicit representation of \(\mathcal{F}_{n}^{ ( \alpha ) } (x;\lambda; \mu;\nu )\) in terms of the Gauss hypergeometric function \({{}_{2}F_{1}}(a,b;c;z)\). We study some relations between the family of Apostol-type polynomials \({\mathcal{F}}_{n}^{(\alpha)}(x;\lambda;\mu;\nu)\) and the family of Hurwitz zeta functions \(\Phi_{\mu} (z,s,a)\). Some special cases also are shown.

2 Multiplication formula for the Apostol-type polynomials

In this section we give a unified multiplication formula for the Apostol-type polynomials \(\mathcal{F}_{n}^{ (\alpha ) } (x;\lambda; \mu;\nu )\). We will see that some well-known results are the corresponding special cases of our result.

First we need the following lemmas.

Lemma 2.1

(Multinomial identity [23], p.28, Theorem B)

If \(x_{1}, x_{2}, \ldots, x_{m}\) are commuting elements of a ring (\(\iff x_{i} x_{j} =x_{j} x_{i}\), \(1\le i < j\le m\)), then we have for all integers \(n\ge 0\):

$$ (x_{1}+x_{2}+\cdots+x_{m})^{n}= \sum_{\substack{a_{1},a_{2},\ldots,a_{m}\ge0 \\ a_{1}+a_{2}+\cdots+a_{m}=n}} {n \choose a_{1}, a_{2},\ldots,a_{m} } {x_{1}}^{a_{1}}{x_{2}}^{a_{2}} \cdots{x_{m}}^{a_{m}}, $$
(2.1)

the last summation takes place over all positive or zero integers \(a_{i}\ge0\) such that \(a_{1}+a_{2}+\cdots+a_{m}=n\), where

$${n \choose a_{1}, a_{2},\ldots,a_{m}}:= \frac{n!}{a_{1}!a_{2}!\cdots a_{m}!}, $$

are called multinomial coefficients defined by [23], p.28, Definition B.

Lemma 2.2

(Generalized multinomial identity [23], p.41, Eq. (12m))

If \(x_{1}, x_{2}, \ldots, x_{m}\) are commuting elements of a ring (\(\iff x_{i} x_{j} =x_{j} x_{i}\), \(1\le i < j\le m\)), then we have for all real or complex variable α:

$$ (1+x_{1}+x_{2}+\cdots+x_{m})^{\alpha}= \sum_{v_{1},v_{2},\ldots,v_{m}\ge0 } {\alpha\choose v_{1}, v_{2},\ldots,v_{m} } {x_{1}}^{v_{1}}{x_{2}}^{v_{2}} \cdots {x_{m}}^{v_{m}}, $$
(2.2)

the last summation takes place over all positive or zero integers \(v_{i}\ge0\), where

$${\alpha\choose v_{1}, v_{2},\ldots,v_{m}}:= \frac{\{\alpha\} _{v_{1}+v_{2}+\cdots+v_{m}}}{v_{1}!v_{2}!\cdots v_{m}!} =\frac{\alpha(\alpha-1)(\alpha-2)\cdots(\alpha -v_{1}-v_{2}-\cdots-v_{m}+1)}{v_{1}!v_{2}!\cdots v_{m}!} $$

are called generalized multinomial coefficients defined by [23], p.27, Eq. (10 \(C^{\prime\prime}\)).

Theorem 2.3

(Multiplication formula)

For \(\mu,\nu, r \in\mathbb{N}\) and \(\nu\leq1\), \(n,l\in{\mathbb {N}}_{0}\), \(\alpha, \lambda \in\mathbb{C}\), we have

$$\begin{aligned}& {\mathcal{F}}_{n}^{(\alpha)}(rx; \lambda; \mu; \nu)=r^{n-\nu {\alpha}} \sum_{v_{1}, v_{2}, \ldots, v_{r-1}\ge0}{\alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}} \\& \hphantom{{\mathcal{F}}_{n}^{(\alpha)}(rx; \lambda; \mu; \nu)=}{}\times(- \lambda)^{m}{\mathcal{F}}_{n}^{(\alpha)} \biggl(x+ \frac{m}{r}; {\lambda}^{r}; \mu;\nu \biggr),\quad r \textit{ odd}, \end{aligned}$$
(2.3)
$$\begin{aligned}& {\mathcal{F}}_{n}^{(l)}(rx; \lambda; \mu;\nu)= \frac{(-1)^{l} 2^{\mu{l}}r^{n-\nu{l}}}{(n+1)_{(1-\nu)l}}\sum_{\substack{0 \le v_{1}, v_{2}, \ldots, v_{r-1}\le l \\ v_{1}+v_{2}+\cdots+v_{r-1}=l}}{l \choose v_{1}, v_{2}, \ldots, v_{r-1}} \\& \hphantom{{\mathcal{F}}_{n}^{(l)}(rx; \lambda; \mu;\nu)=}{}\times (- \lambda)^{m}{\mathcal{B}}_{n+(1-\nu)l}^{(l)} \biggl(x+ \frac{m}{r}; {\lambda}^{r} \biggr), \quad r \textit{ even}, \end{aligned}$$
(2.4)

where \(m=v_{1}+2v_{2}+\cdots+(r-1)v_{r-1}\).

Proof

It is not difficult to show that

$$ \frac{1}{\lambda e^{z} +1}=-\frac{1-\lambda e^{z}+{\lambda}^{2}e^{2z}+\cdots+{(-\lambda )}^{r-1}e^{(r-1)z}}{{(-\lambda)}^{r}e^{rz}-1}. $$
(2.5)

When r is odd, by (1.18) and (2.5) we get

$$\begin{aligned}& \sum_{n=0}^{\infty}{\mathcal {F}}_{n}^{(\alpha)}(rx; \lambda;\mu;\nu)\frac{z^{n}}{n!} \\& \quad = \frac {1}{r^{\nu{\alpha}}} \biggl(\frac{2^{\mu}(rz)^{\nu}}{ {\lambda}^{r}e^{rz}+1} \biggr)^{\alpha} \biggl( \frac{{\lambda}^{r}e^{rz}+1}{ \lambda e^{z}+1} \biggr)^{\alpha}e^{rxz} \\& \quad = \frac{1}{r^{\nu{\alpha}}} \biggl(\frac{2^{\mu}(rz)^{\nu}}{ {\lambda}^{r}e^{rz}+1} \biggr)^{\alpha} \Biggl( \sum_{k=0}^{r-1}\bigl(-\lambda{e^{z}} \bigr)^{k} \Biggr)^{\alpha}e^{rxz} \\& \quad = \frac{1}{r^{\nu{\alpha}}}\sum_{v_{1}, v_{2},\ldots,v_{r-1}\ge 0}{\alpha \choose v_{1}, v_{2}, \ldots, v_{r-1}}(- \lambda)^{m} \biggl(\frac{2^{\mu }(rz)^{\nu}}{ {\lambda}^{r}e^{rz}+1} \biggr)^{\alpha}e^{(x+\frac{m}{r})rz} \\& \quad = \sum_{n=0}^{\infty} \biggl[r^{n-\nu{\alpha}} \sum_{v_{1},v_{2},\ldots ,v_{r-1}\ge 0}{\alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}}(-\lambda)^{m}{ \mathcal{F}}_{n}^{(\alpha)} \biggl(x+\frac {m}{r};{ \lambda}^{r};\mu;\nu \biggr) \biggr]\frac{z^{n}}{n!}. \end{aligned}$$
(2.6)

Comparing the coefficients of \(\frac{z^{n}}{n!}\) on both sides of (2.6), we obtain the assertion (2.3) of Theorem 2.3.

When r is even, we can similarly prove the assertion (2.4) of Theorem 2.3. The proof is complete. □

It follows that we can deduce the well-known formulas from Theorem 2.3.

Letting \(\lambda\longmapsto-\lambda\), taking \(\mu=0\) and \(\nu=1\) in (2.3) and (2.4) and noting (1.19), we can obtain the following main result of Luo (see [24], p.380, Theorem 2.1).

Corollary 2.4

For \(r, \alpha\in\mathbb{N}\), \(n \in{\mathbb{N}}_{0}\), \(\lambda\in \mathbb{C}\), the following multiplication formula for the Apostol-Bernoulli polynomials of higher order holds true:

$$ {\mathcal{B}}_{n}^{(\alpha)}(rx; \lambda)=r^{n-\alpha} \sum _{v_{1}, v_{2}, \ldots, v_{r-1}\ge0}{\alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}} {\lambda}^{m}{ \mathcal{B}}_{n}^{(\alpha)} \biggl(x+\frac{m}{r}; {\lambda }^{r} \biggr), $$
(2.7)

where \(m=v_{1}+2v_{2}+\cdots+(r-1)v_{r-1}\).

Taking \(\mu=1\) and \(\nu=0\) in (2.3) and (2.4), and noting (1.20), we can obtain the following main result of Luo (see [24], p.385, Theorem 3.1).

Corollary 2.5

For \(r \in\mathbb{N}\), \(n,l\in{\mathbb{N}}_{0}\), \(\alpha, \lambda \in\mathbb{C}\), the following multiplication formula for the Apostol-Euler polynomials of higher order holds true:

$$\begin{aligned}& {\mathcal{E}}_{n}^{(\alpha)}(rx; \lambda)=r^{n} \sum _{v_{1}, v_{2}, \ldots, v_{r-1}\ge0}{\alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}} (-\lambda)^{m}{ \mathcal{E}}_{n}^{(\alpha)} \biggl(x+\frac{m}{r}; { \lambda}^{r} \biggr), \quad r \textit{ odd}, \end{aligned}$$
(2.8)
$$\begin{aligned}& {\mathcal{E}}_{n}^{(l)}(rx; \lambda)=\frac{(-2)^{l} r^{n}}{(n+1)_{l}}\sum _{\substack{0 \le v_{1}, v_{2}, \ldots, v_{r-1}\le l \\ v_{1}+v_{2}+\cdots+v_{r-1}=l}}{l \choose v_{1}, v_{2}, \ldots, v_{r-1}} \\& \hphantom{{\mathcal{E}}_{n}^{(l)}(rx; \lambda)=}{}\times(-\lambda)^{m}{\mathcal{B}}_{n+l}^{(l)} \biggl(x+\frac{m}{r}; {\lambda }^{r} \biggr), \quad r \textit{ even}, \end{aligned}$$
(2.9)

where \(m=v_{1}+2v_{2}+\cdots+(r-1)v_{r-1}\).

Taking \(\mu=\nu=1\) in (2.3) and (2.4), and noting (1.21), we can obtain the following main result (see [14], p.2462, Corollary 4.6).

Corollary 2.6

For \(\alpha, r \in\mathbb{N}\), \(n,l\in{\mathbb{N}}_{0}\), \(\lambda \in\mathbb{C}\), the following multiplication formula for the Apostol-Genocchi polynomials of higher order holds true:

$$\begin{aligned}& {\mathcal{G}}_{n}^{(\alpha)}(rx; \lambda)=r^{n-\alpha} \sum _{v_{1}, v_{2}, \ldots, v_{r-1}\ge0}{\alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}} (-\lambda)^{m}{ \mathcal{G}}_{n}^{(\alpha)} \biggl(x+\frac{m}{r}; { \lambda}^{r} \biggr), \quad r \textit{ odd}, \end{aligned}$$
(2.10)
$$\begin{aligned}& {\mathcal{G}}_{n}^{(l)}(rx; \lambda)=(-2)^{l} r^{n-l}\sum_{\substack{0 \le v_{1}, v_{2}, \ldots, v_{r-1}\le l \\ v_{1}+v_{2}+\cdots+v_{r-1}=l}}{l \choose v_{1}, v_{2}, \ldots, v_{r-1}} \\& \hphantom{{\mathcal{G}}_{n}^{(l)}(rx; \lambda)=}{}\times(- \lambda)^{m}{\mathcal{B}}_{n}^{(l)} \biggl(x+ \frac{m}{r}; {\lambda}^{r} \biggr), \quad r \textit{ even}, \end{aligned}$$
(2.11)

where \(m=v_{1}+2v_{2}+\cdots+(r-1)v_{r-1}\).

Taking \(\lambda=- (\frac{\beta}{a} )^{b}\), \(\mu=1-\kappa\), \(\nu=\kappa\) in (2.3), and noting (1.23), we can obtain the following multiplication formulas for the polynomials \({\mathcal{Y}}^{(\alpha)}_{n,\beta}(x;\kappa,a,b)\) and \(\mathcal {Y}_{n,\beta} (x;\kappa,a,b )\) defined by (1.16) and (1.17), respectively.

Corollary 2.7

For \(\kappa, \mu, \nu, m, n, l, r \in{\mathbb{N}}_{0}\), \(\alpha, \lambda \in\mathbb{C}\), we have

$$\begin{aligned}& {\mathcal{Y}}^{(\alpha)}_{n,\beta}(rx;\kappa,a,b) \\& \quad =r^{n-\kappa{\alpha}} \sum_{v_{1}, v_{2}, \ldots, v_{r-1}\ge0}{\alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}} \biggl( \frac{\beta}{a} \biggr)^{bm}a^{(r-1)b{\alpha }}{\mathcal{Y}}_{n,\beta}^{(\alpha)} \biggl(x+\frac{m}{r}; \kappa;a;br \biggr) \end{aligned}$$
(2.12)
$$\begin{aligned}& \quad =r^{n-\kappa{\alpha}} \sum_{v_{1}, v_{2}, \ldots, v_{r-1}\ge0}{ \alpha\choose v_{1}, v_{2}, \ldots, v_{r-1}} \biggl(\frac{\beta}{a} \biggr)^{bm}a^{(r-1)b{\alpha }}{ \mathcal{Y}}_{n,\beta^{r}}^{(\alpha)} \biggl(x+\frac{m}{r}; \kappa;a^{r};b \biggr), \end{aligned}$$
(2.13)

where \(m=v_{1}+2v_{2}+\cdots+(r-1)v_{r-1}\).

Setting \(\alpha=l=1\) in (2.12) and (2.13), respectively, we have (see [13], p.2786, Theorem 8) the following.

Corollary 2.8

For \(\kappa, \mu, \nu, n , r \in{\mathbb{N}}_{0}\), \(\lambda \in\mathbb{C}\), we have

$$\begin{aligned} {\mathcal{Y}}_{n,\beta}(rx;\kappa,a,b)&= r^{n-\kappa} \sum _{j=0}^{r-1} \biggl(\frac{\beta}{a} \biggr)^{bj}a^{(r-1)b}{\mathcal {Y}}_{n,\beta} \biggl(x+ \frac{j}{r}; \kappa;a;br \biggr) \end{aligned}$$
(2.14)
$$\begin{aligned} &= r^{n-\kappa} \sum_{j=0}^{r-1} \biggl(\frac{\beta}{a} \biggr)^{bj}a^{(r-1)b}{ \mathcal{Y}}_{n,\beta^{r}} \biggl(x+\frac{j}{r}; \kappa;a^{r};b \biggr). \end{aligned}$$
(2.15)

Remark 2.9

In [14], p.2460, Theorem 4.3, one of the main result of Özarslan is not right, the correct form should be (2.12) and (2.13) of Corollary 2.7.

Remark 2.10

In fact, setting \(\lambda=- (\frac{\beta}{a} )^{b}\), \(\mu=1-\kappa\), \(\nu=\kappa\) in (2.3) and noting (1.23), we deduce the multiplication formulas which are right only when r is odd. In the same way as the proof of [24], p.380, Theorem 2.1, we can obtain the multiplication formulas (2.12) and (2.13) of Corollary 2.7.

3 A unified representation in conjunction with the Gauss hypergeometric function

In this section we obtain a unified representation of the Apostol-type polynomials \({\mathcal{F}}_{n}^{(l)}(x;\lambda;\mu;\nu)\) with the Gaussian hypergeometric functions.

Theorem 3.1

For \(\mu,\nu,n,l \in{\mathbb{N}}_{0}\), \(\lambda\in\mathbb{C}\), we have

$$\begin{aligned}& {\mathcal{F}}_{n}^{(l)}(x;\lambda;\mu; \nu) \\& \quad = 2^{\mu{l}}(\nu{l})!{n \choose \nu{l}}\sum _{k=0}^{n-\nu{l}}{l+k-1 \choose k} {n-\nu{l} \choose k} \frac{(-\lambda)^{k}}{(\lambda+1)^{l+k}} \\& \qquad {} \times\sum_{m=0}^{k}(-1)^{m}{k \choose m}m^{k} (x+m)^{n-\nu{l}-k}{ {}_{2}F_{1}} \biggl(-n+\nu{l}+k,k;k+1;\frac {m}{m+x} \biggr), \end{aligned}$$
(3.1)

where \(F(a,b;c;z)\) denotes Gaussian hypergeometric functions defined by (see [2], p.44, Eq. (4))

$$ F(a,b;c;z):=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!},\quad \vert z\vert < 1. $$
(3.2)

Proof

Letting \(\alpha=l \in{\mathbb{N}}\) in (1.18), we have

$$ \sum_{n=0}^{\infty}{\mathcal {F}}_{n}^{(l)}(x;\lambda;\mu;\nu)\frac{z^{n}}{n!}= { \biggl( \frac {2^{\mu}z^{\nu}}{\lambda e^{z}+1} \biggr)}^{l} e^{xz}. $$
(3.3)

Differentiating both sides of (3.3) with respect to the variable z yields

$$\begin{aligned} {\mathcal{F}}_{n}^{(l)}(x;\lambda;\mu;\nu)&=D_{z}^{n} \biggl[ \biggl(\frac {2^{\mu}z^{\nu}}{\lambda e^{z}+1} \biggr)^{l}e^{xz} \biggr]_{z=0} \\ &=2^{\mu{l}}\sum_{s=0}^{n}{n \choose s}x^{n-s}D_{z}^{s} \bigl[z^{\nu {l}}\bigl( \lambda e^{z}+1\bigr)^{-l} \bigr]_{z=0} \\ &=2^{\mu{l}}\sum_{s=\nu{l}}^{n}{n \choose s}x^{n-s}(\nu{l})!{s \choose \nu{l}}D_{z}^{s-\nu{l}} \bigl[\bigl(\lambda e^{z}+1\bigr)^{-l} \bigr]_{z=0} \\ &=2^{\mu{l}}\sum_{s=\nu{l}}^{n}{n \choose s}x^{n-s}(\nu{l})!{s \choose \nu{l}}D_{z}^{s-\nu{l}} \bigl[ \bigl(\lambda+1+\lambda \bigl(e^{z}-1\bigr) \bigr)^{-l} \bigr]_{z=0}, \end{aligned}$$

where \(D_{z}=\frac{d}{dz}\) is the differential operator.

Applying the generalized binomial theorem

$$(a+b)^{-\alpha}=\sum_{l=0}^{\infty}{ \alpha+l-1 \choose l}a^{-\alpha -l}(-b)^{l} \quad \biggl(\alpha\in \mathbb{C}, \biggl\vert \frac {b}{a}\biggr\vert < 1 \biggr) $$

and the generating function of the Stirling numbers of the second kind \(S(n,k)\) (see, for details, [23], p.206, Theorem A),

$$\frac{(e^{z}-1)^{k}}{k!}=\sum_{n=0}^{\infty}S(n,k) \frac{z^{n}}{n!}, $$

we find that

$$\begin{aligned}& {\mathcal{F}}_{n}^{(l)}(x;\lambda;\mu;\nu) \\& \quad =2^{\mu{l}}\sum_{s=\nu{l}}^{n}{n \choose s}x^{n-s}(\nu{l})!{s \choose \nu{l}}\sum _{k=0}^{\infty}{l+k-1 \choose k}(\lambda+1)^{-l-k}(- \lambda)^{k}D_{z}^{s-\nu{l}}\bigl[\bigl(e^{z}-1 \bigr)^{k}\bigr]_{z=0} \\& \quad =2^{\mu{l}}\sum_{s=\nu{l}}^{n}{n \choose s}x^{n-s}(\nu{l})!{s \choose \nu{l}}\sum _{k=0}^{s-\nu{l}}{l+k-1 \choose k}(-\lambda)^{k}( \lambda +1)^{-l-k}k!S(s-\nu{l},k). \end{aligned}$$

Noting (see [2], p.58, Eq. (20))

$$ S(n,k)=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j} \binom{k}{j}j^{n} $$

and the well-known combinatorial identity

$$ \binom{n}{k}\binom{k}{s}=\binom{n}{s}\binom{n-s}{n-k}, $$

we readily obtain

$$\begin{aligned}& {\mathcal{F}}_{n}^{(l)}(x;\lambda;\mu;\nu) \\& \quad = 2^{\mu{l}}\sum_{s=\nu{l}}^{n}{n \choose s}x^{n-s}(\nu{l})!{s \choose \nu{l}}\sum_{k=0}^{s-\nu{l}}{l+k-1 \choose k} \\& \qquad {}\times(-\lambda)^{k}(\lambda +1)^{-l-k}\sum _{m=0}^{k}(-1)^{k-m}{k \choose m}m^{s-\nu{l}} \\& \quad = 2^{\mu{l}}(\nu{l})!{n \choose \nu{l}}\sum_{k=0}^{n-\nu{l}} \sum_{s=k+\nu{l}}^{n}{n-\nu{l} \choose n-s} {l+k-1 \choose k}\frac{(-\lambda)^{k}x^{n-s}}{(\lambda +1)^{l+k}}\sum_{m=0}^{k}(-1)^{k-m}{k \choose m}m^{s-\nu{l}} \\& \quad = 2^{\mu{l}}(\nu{l})!{n \choose \nu{l}}\sum_{k=0}^{n-\nu{l}} \sum_{s=0}^{n-k-\nu{l}}{n-\nu{l} \choose n-s- \nu{l}-k} {l+k-1 \choose k} \\& \qquad {}\times\frac{(-\lambda)^{k}x^{n-s-k-\nu{l}}}{(\lambda +1)^{l+k}}\sum_{m=0}^{k}(-1)^{k-m}{k \choose m}m^{s+k} \\& \quad = 2^{\mu{l}}(\nu{l})!{n \choose \nu{l}}\sum_{k=0}^{n-\nu{l}}{l+k-1 \choose k}\frac{(-\lambda)^{k}x^{n-k-\nu{l}}}{(\lambda +1)^{l+k}}\sum_{m=0}^{k}(-1)^{k-m}{k \choose m}m^{k} \\& \qquad {}\times\sum_{s=0}^{n-k-\nu {l}}{n- \nu{l} \choose n-s-\nu{l}-k} \biggl(\frac{m}{x} \biggr)^{s}. \end{aligned}$$

Noting that (in view of \(\binom{n}{k}=0\) when \(k>n\) or \(k<0\))

$$\sum_{k=0}^{n}\binom{n}{k}=\sum _{k=0}^{\infty}\binom{n}{k}, $$

and combining the definition of the Gaussian hypergeometric function

$$ {}_{2}F_{1}(a,b;c;z):= \sum _{n=0}^{\infty}\frac {(a)_{n}(b)_{n}}{(c)_{n}} \frac{z^{n}}{n!}, $$

we obtain

$$\begin{aligned} {\mathcal{F}}_{n}^{(l)}(x;\lambda;\mu;\nu) =&2^{\mu{l}}(\nu{l})!{n \choose \nu{l}}\sum_{k=0}^{n-\nu{l}}{l+k-1 \choose k} {n-\nu{l} \choose k} \\ &{}\times\frac{(-\lambda)^{k}x^{n-k-\nu{l}}}{(\lambda+1)^{l+k}}\sum _{m=0}^{k}(-1)^{m}{k \choose m}m^{k} \\ &{} \times{ {}_{2}F_{1}} \biggl(-n+\nu{l}+k,1;k+1;- \frac {m}{x} \biggr). \end{aligned}$$
(3.4)

Applying the Pfaff-Kummer hypergeometric transformation [25], p.559, Eq. (15.3.4),

$$ { {}_{2}F_{1}}(a,b;c;z)=(1-z)^{-a}{ {}_{2}F_{1}} \biggl(a,c-b;c;\frac {z}{z-1} \biggr)\quad \bigl(c \notin{\mathbb{Z}}_{0}^{-}:\bigl\vert \operatorname{arg}(1-z) \bigr\vert \leq\pi -\epsilon\ (0< \epsilon <\pi)\bigr), $$

to (3.4), we arrive at the desired equation, (3.1). This completes our proof. □

Below we show some special cases of (3.1).

Letting \(\lambda\longmapsto-\lambda\), taking \(\mu=0\) and \(\nu=1\) in (3.1) and noting (1.19), we easily obtain the following explicit formula for the Apostol-Bernoulli polynomials:

$$\begin{aligned} {\mathcal{B}}_{n}^{(l)}(x;\lambda) =&l!{n \choose l}\sum _{k=0}^{n-l}{l+k-1 \choose k} {n-l \choose k} \frac{{\lambda}^{k}}{(\lambda-1)^{k}} \\ &{}\times\sum_{m=0}^{k}(-1)^{m}{k \choose m}m^{k} (x+m)^{n-l-k}{ {}_{2}F_{1}} \biggl(-n+l+k,k;k+1;\frac{m}{m+x} \biggr), \end{aligned}$$
(3.5)

with \(n, l \in{\mathbb{N}}_{0}\), \(\lambda\in\mathbb{C}\setminus\{1\} \), which is just the main result of Luo and Srivastava (see [5], p.294, Theorem 1).

Taking \(\mu=1\) and \(\nu=0\) in (3.1) and noting (1.20), we can obtain the following explicit formula for the Apostol-Euler polynomials:

$$\begin{aligned} {\mathcal{E}}_{n}^{(l)}(x;\lambda) =&2^{l}\sum _{k=0}^{n}{l+k-1 \choose k} {n \choose k} \frac{(-\lambda)^{k}}{(\lambda+1)^{l+k}} \\ &{}\times\sum_{m=0}^{k}(-1)^{m}{k \choose m}m^{k} (x+m)^{n-k}{ {}_{2}F_{1}} \biggl(-n+k,k;k+1;\frac{m}{m+x} \biggr), \end{aligned}$$
(3.6)

with \(n,l\in{\mathbb{N}}_{0}\), \(\lambda\in\mathbb{C}\setminus\{-1\} \), which is just the main result of Luo (see [6], p.920, Theorem 1).

Taking \(\mu=\nu=1\) in (3.1), and noting (1.21), we can obtain the following explicit representation of the generalized Apostol-Genocchi polynomials:

$$\begin{aligned} {\mathcal{G}}_{n}^{(l)}(x;\lambda) =&2^{l}l!{n \choose l}\sum_{k=0}^{n-l}{l+k-1 \choose k} {n-l \choose k}\frac{(-\lambda)^{k}}{(\lambda+1)^{l+k}} \\ &{}\times\sum_{m=0}^{k}(-1)^{m}{k \choose m}m^{k} (x+m)^{n-l-k}{ {}_{2}F_{1}} \biggl(-n+l+k,k;k+1;\frac{m}{m+x} \biggr), \end{aligned}$$
(3.7)

with \(n,l\in{\mathbb{N}}_{0}\), \(\lambda\in\mathbb{C}\setminus\{-1\} \), which is just one of the results of Luo and Srivastava (see [15], p.5708, Theorem 1).

Taking \(\lambda=- (\frac{\beta}{a} )^{b}\), \(\mu=1-\kappa\), \(\nu=\kappa\) in (3.1), and noting (1.23), we deduce the following well-known formula:

$$\begin{aligned} {\mathcal{Y}}^{(l)}_{n,\beta}(x; \kappa,a,b) =&2^{l(1-\kappa)}(l \kappa)! \binom{l+k-1}{k}{n \choose l \kappa} \sum_{k=0}^{n-l \kappa}{n-l \kappa\choose k} \frac{{\beta}^{bk}}{({\beta}^{b}-a^{b})^{k+1}} \\ &{}\times\sum_{m=0}^{k}(-1)^{m}{k \choose m} m^{k}(x+m)^{n-k-l \kappa} \\ &{}\times{ {}_{2}F_{1}} \biggl(-n+l \kappa+k,k;k+1;\frac {m}{m+x} \biggr), \end{aligned}$$
(3.8)

with \(n,l,\kappa\in{\mathbb{N}}_{0}\), \(\beta\in\mathbb{C}\), \(a,b \in \mathbb{C}\setminus\{0\}\), \(\beta\neq a\), which is just main result of Özarslan (see [14], p.2454, Theorem 2.1).

Further setting \(l=1\) in (3.8) we deduce the following formula for \({\mathcal{Y}}_{n,\beta}(x;\kappa,a,b)\):

$$\begin{aligned} {\mathcal{Y}}_{n,\beta}(x;\kappa,a,b) =&2^{1-\kappa}\kappa!{n \choose \kappa}\sum_{k=0}^{n-\kappa}{n-\kappa\choose k} \frac{{\beta}^{bk}}{({\beta}^{b}-a^{b})^{k+1}} \\ &{}\times\sum_{m=0}^{k}(-1)^{m}{k \choose m} m^{k}(x+m)^{n-k-\kappa} \\ &{}\times{ {}_{2}F_{1}} \biggl(-n+\kappa+k,k;k+1;\frac {m}{m+x} \biggr). \end{aligned}$$
(3.9)

4 Some explicit relationships between the generalized Apostol-type polynomials and generalized Hurwitz-Lerch zeta function

A general Hurwitz-Lerch zeta function \(\Phi(z,s,a)\) defined by (cf., e.g., [2], p.121, et seq.)

$$\begin{aligned}& \Phi(z,s,a):=\sum_{n=0}^{\infty} \frac{z^{n}}{(n+a)^{s}} \\& \quad \bigl( a\in\mathbb{C} \setminus\mathbb{Z}_{0}^{-}; s\in \mathbb{C}\text{ when }\vert z\vert < 1; \mathfrak{R}(s)>1\text{ when }\vert z\vert =1 \bigr) \end{aligned}$$
(4.1)

contains, as special cases, not only the Hurwitz (or generalized) zeta function \(\zeta(s,a)\) defined by (cf. [26], p.249 and [4], p.88)

$$ \zeta(s,a):=\Phi(1,s,a)=\sum_{n=0}^{\infty} \frac{1}{(n+a)^{s}} \quad \bigl(\Re(s)>1;a \in\notin\mathbb{Z}_{0}^{-} \bigr) $$
(4.2)

and the Riemann zeta function \(\zeta(s)\),

$$ \zeta(s):=\Phi(1,s,1)=\zeta(s,1)=\frac{1}{2^{s}-1}\zeta \biggl(s, \frac{1}{2} \biggr) \quad \bigl(\Re(s)>1;a \notin\mathbb{Z}_{0}^{-} \bigr), $$
(4.3)

and the Lerch zeta function:

$$ \ell_{s}(\xi):=\sum_{n=1}^{\infty} \frac{e^{2n\pi i\xi }}{n^{s}}=e^{2\pi i\xi} \Phi \bigl( e^{2\pi i\xi},s,1 \bigr)\quad \bigl( \xi\in\mathbb{R}; \mathfrak{R}(s)>1 \bigr), $$
(4.4)

but also such other functions as the polylogarithm function:

$$\begin{aligned}& \operatorname{Li}_{s}(z):=\sum_{n=1}^{\infty} \frac{z^{n}}{n^{s}}=z \Phi (z,s,1) \\& \quad \bigl(s\in\mathbb{C }\text{ when } \vert z \vert < 1; \mathfrak{R}(s)>1 \text{ when } \vert z\vert =1 \bigr) \end{aligned}$$
(4.5)

and the Lipschitz-Lerch zeta function (cf. [2], p.122, Eq. 2.5(11)):

$$\begin{aligned}& \phi(\xi,a,s):=\sum_{n=0}^{\infty} \frac{e^{2n\pi i\xi }}{(n+a)^{s}}=\Phi \bigl( e^{2\pi i\xi},s,a \bigr) =:L ( \xi,s,a ) \\& \quad \bigl(a\in\mathbb{C} \setminus\mathbb{Z}_{0}^{-}; \mathfrak{R}(s)>0 \text{ when }\xi\in\mathbb{R }\setminus \mathbb{Z}; \mathfrak{R}(s)>1 \text{ when }\xi\in \mathbb{Z} \bigr), \end{aligned}$$
(4.6)

which was first studied by Rudolf Lipschitz (1832-1903) and Matyáš Lerch (1860-1922) in connection with Dirichlet’s famous theorem on primes in arithmetic progressions.

A family of the Hurwitz-Lerch zeta functions \(\Phi_{\mu,\nu}^{(\rho ,\sigma)} (z,s,a)\) defined by (see e.g. [27], p.727, Eq. (8))

$$\begin{aligned}& \Phi_{\mu,\nu}^{(\rho,\sigma)} (z,s,a):=\sum _{n=0}^{\infty}\frac {(\mu)_{\rho{n}}}{(\nu)_{\sigma{n}}}\frac{z^{n}}{(n+a)^{s}} \\& \quad \bigl( \mu\in\mathbb{C}; a, \nu\in\mathbb{C}\setminus \mathbb{Z}_{0}^{-}; \rho, \sigma\in\mathbb{R}^{+}; \rho< \sigma \text{ when } s, z \in\mathbb{C}; \\& \quad \rho =\sigma \text{ and } s \in\mathbb{C} \text{ when } \vert z \vert<1; \rho= \sigma \text{ and } \Re(s-\mu+\nu)>1 \text { when } \vert z \vert= 1\bigr), \end{aligned}$$
(4.7)

contains, as special cases, not only the Hurwitz-Lerch zeta function

$$ \Phi_{\nu,\nu}^{(\sigma,\sigma)} (z,s,a)=\Phi_{\mu,\nu}^{(0,0)} (z,s,a)=\Phi(z,s,a)=\sum_{n=0}^{\infty} \frac{z^{n}}{(n+a)^{s}} $$
(4.8)

and the Lipschitz-Lerch zeta function \(\phi(\xi,a,s):=\Phi ( e^{2\pi i\xi},s,a )\), but also the following generalized Hurwitz-Lerch zeta functions introduced and studied earlier by Goyal and Laddha [28], p.100, Eq. (1.5):

$$ \Phi_{\mu,1}^{(1,1)} (z,s,a)=\Phi_{\mu} (z,s,a):=\sum_{n=0}^{\infty}\frac{(\mu)_{n}}{n!} \frac{z^{n}}{(n+a)^{s}}, $$
(4.9)

which, are called the Goyal-Laddha-Hurwitz-Lerch zeta functions.

Below we give an explicit relationship between the Apostol-type polynomials \({\mathcal{F}}_{n}^{(\alpha)}(x;\lambda; \mu;\nu)\) and the Hurwitz-Lerch zeta function \(\Phi_{\mu} (z,s,a)\).

Theorem 4.1

For \(n, \nu, \alpha\in\mathbb{N}_{0}\); \(-1<\lambda\leqq1\); \(n \geqq\nu\alpha\); \(x \in\mathbb{C} \setminus \mathbb{Z}_{0}^{-}\), \(\mu\in\mathbb{C}\), the relationship

$$ {\mathcal {F}}_{n}^{(\alpha)}(x;\lambda;\mu;\nu) =2^{\mu\alpha}(\nu\alpha)! \binom{n}{\nu\alpha} \Phi_{\alpha}(-\lambda,\nu \alpha-n,x) $$
(4.10)

holds true.

Proof

Applying the generalized binomial theorem

$$ (1+w)^{-\alpha}=\sum_{r=0}^{\infty} \binom{\alpha +r-1}{r}(-w)^{r} \quad \bigl( \vert w\vert < 1 \bigr) $$

in (1.18), we have

$$\begin{aligned} \sum_{n=0}^{\infty}{\mathcal {F}}_{n}^{(\alpha)}(x;\lambda;\mu;\nu)\frac{z^{n}}{n!} &={ \biggl(\frac{2^{\mu}z^{\nu}}{\lambda e^{z}+1} \biggr)}^{\alpha} e^{xz} \\ &= 2^{\mu\alpha}z^{\nu\alpha} \bigl(1 +\lambda e^{z} \bigr)^{-\alpha} e^{xz} \\ &= 2^{\mu\alpha}z^{\nu\alpha}\sum_{k=0}^{\infty} \frac{(\alpha )_{k}}{k!} (-\lambda)^{k} e^{(k+x)z} \\ &= \sum_{n=0}^{\infty} \Biggl[2^{\mu\alpha} \sum_{k=0}^{\infty }\frac{(\alpha)_{k}}{k!} (- \lambda)^{k} (k+x)^{n} \Biggr] \frac {z^{n+\nu\alpha}}{n!} \\ &= \sum_{n=\nu\alpha}^{\infty} \Biggl[2^{\mu\alpha}( \nu\alpha)! \binom{n}{\nu\alpha}\sum_{k=0}^{\infty} \frac{(\alpha)_{k}}{k!} \frac{(-\lambda)^{k}}{(k+x)^{\nu\alpha-n}} \Biggr] \frac{z^{n}}{n!}. \end{aligned}$$
(4.11)

Noting (4.9), (4.10) follows. □

Below we see that (4.10) implies some well-known results.

Let \(\lambda\longmapsto-\lambda\), taking \(\mu=0\) and \(\nu=1\) in (4.10) and noting (1.19), we can obtain an explicit relation between the Apostol-Bernoulli polynomials and the Hurwitz-Lerch zeta function:

$$ \mathcal{B}_{n}^{ ( l ) } (x;\lambda ) = (-n)_{l} \Phi_{l}(\lambda,l-n,x)\quad \bigl(n, l \in\mathbb{N}; n \geqq l; \vert \lambda \vert < 1; x \in\mathbb{C} \setminus\mathbb{Z}_{0}^{-} \bigr). $$
(4.12)

The above result is just one of the main results of Garg et al. (see [29], p.809).

Clearly, we have the following relation between the Apostol-Bernoulli polynomials and the Hurwitz-Lerch zeta function:

$$ \mathcal{B}_{n} (x;\lambda ) = -n \Phi(\lambda,1-n,x) \quad \bigl(n \in\mathbb{N}; \vert \lambda \vert \leqq1; x \in\mathbb{C} \setminus\mathbb{Z}_{0}^{-}\bigr), $$
(4.13)

which is just the result of Apostol (see [3]).

Taking \(\lambda=1\) in (4.13), we obtain the following well-known relationship between the Bernoulli polynomials and Hurwitz zeta function (see [26], p.264, Theorem 12.13):

$$ B_{n}(x) =- n \zeta(1-n,x) \quad (n \in\mathbb{N}). $$
(4.14)

Taking \(x=0\) in (4.14), we obtain the following the well-known relationship between the Bernoulli numbers and the Riemann zeta function (see [26], p.266, Theorem 12.16):

$$ B_{n} =- n \zeta(1-n)\quad (n \in\mathbb{N}). $$
(4.15)

Taking \(\mu=1\) and \(\nu=0\) in (4.10) and noting (1.20), we can obtain the following result of Luo (see [30], p.339, Theorem 2.1):

$$ \mathcal{E}_{n}^{(\alpha)} (x;\lambda ) =2^{\alpha}\Phi _{\alpha}(-\lambda,-n,x) . $$
(4.16)

Further taking \(\alpha=1\) in (4.16), we have the following relation between the Apostol-Euler polynomials and the Hurwitz-Lerch zeta function:

$$ \mathcal{E}_{n} (x;\lambda ) = 2 \Phi(-\lambda,-n,x) \quad \bigl(n \in\mathbb{N}; -1< {\lambda} \leqq1; x \in\mathbb{C} \setminus \mathbb{Z}_{0}^{-}\bigr). $$
(4.17)

Taking \(\lambda=1\) in (4.17), we can obtain the following well-known relation between the Euler polynomials and the L-function:

$$ E_{n} (x ) = 2 L(-n,x), $$
(4.18)

where the L-function is defined by

$$ L(s,x):=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{ ( n+x )^{s}}\quad \bigl( \Re ( s ) >1; x \in\mathbb{C} \setminus \mathbb{Z}_{0}^{-} \bigr). $$
(4.19)

From (4.19), we further obtain the following well-known relation between the Euler numbers and the l-function:

$$ E_{n} = 2 l(-n), $$
(4.20)

where the l-function is defined by

$$ l(s):=\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{s}}, \quad \Re(s) >0. $$
(4.21)

Taking \(\mu=\nu=1\) in (4.10), and noting (1.21), we can deduce the following relation between the Apostol-Genocchi polynomials and Hurwitz-Lerch zeta function (see [31], p.124, Corollary 4.2):

$$ \mathcal{G}_{n}^{ ( l ) } (x;\lambda ) = \{n\} _{l} 2^{l} \Phi_{l}(-\lambda,l-n,x)\quad \bigl(n, l \in\mathbb{N}; n \geqq l; \vert \lambda \vert \leqq1; \text { } x \in \mathbb{C} \setminus\mathbb{Z}_{0}^{-}\bigr) $$
(4.22)

and

$$ \mathcal{G}_{n} (x;\lambda ) = 2n \Phi(-\lambda,1-n,x) \quad \bigl(n \in\mathbb{N}; \vert \lambda \vert \leqq1; x \in\mathbb{C} \setminus\mathbb{Z}_{0}^{-}\bigr). $$
(4.23)

Taking \(\lambda=- (\frac{\beta}{a} )^{b}\), \(\mu=1-\kappa\), \(\nu=\kappa\) in (4.10), and noting (1.23), we can deduce the following relations between the polynomials \({\mathcal {Y}}^{(\alpha)}_{n,\beta}(x;\kappa,a,b)\), \({\mathcal{Y}}_{n,\beta }(x;\kappa,a,b)\), and the (generalized) Hurwitz zeta functions [13, 14, 18]:

$$ {\mathcal{Y}}^{(\alpha)}_{n,\beta}(x;\kappa,a,b) =(-1)^{\alpha } \frac{2^{(1-\kappa) \alpha}(\kappa\alpha)!}{a^{b \alpha}} \binom {n}{\kappa\alpha} \Phi_{\alpha} \biggl( \biggl( \frac{\beta}{a} \biggr)^{b},\kappa\alpha-n,x \biggr) $$
(4.24)

and

$$ {\mathcal{Y}}_{n,\beta}(x;\kappa,a,b) =-\frac{2^{(1-\kappa )}(\kappa)!}{a^{b}} \binom{n}{\kappa} \Phi \biggl( \biggl(\frac {\beta}{a} \biggr)^{b}, \kappa-n,x \biggr). $$
(4.25)