Theory and Modern Applications

# On the generalized Apostol-type Frobenius-Euler polynomials

## Abstract

The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz’s results which are associated with Frobenius-Euler polynomials.

MSC:05A10, 11B65, 28B99, 11B68.

## 1 Introduction, definitions and notations

Throughout this presentation, we use the following standard notions: $\mathbb{N}=\left\{1,2,\dots \right\}$, ${\mathbb{N}}_{0}=\left\{0,1,2,\dots \right\}=\mathbb{N}\cup \left\{0\right\}$, ${\mathbb{Z}}^{-}=\left\{-1,-2,\dots \right\}$. Also, as usual denotes the set of integers, denotes the set of real numbers and denotes the set of complex numbers. Furthermore, ${\left(\lambda \right)}_{0}=1$ and

${\left(\lambda \right)}_{k}=\lambda \left(\lambda +1\right)\left(\lambda +2\right)\cdots \left(\lambda +k-1\right),$

where $k\in \mathbb{N}$, $\lambda \in \mathbb{C}$.

The classical Frobenius-Euler polynomial ${H}_{n}^{\left(\alpha \right)}\left(x;u\right)$ of order α is defined by means of the following generating function:

${\left(\frac{1-u}{{e}^{t}-u}\right)}^{\alpha }{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{H}_{n}^{\left(\alpha \right)}\left(x;u\right)\frac{{t}^{n}}{n!},$
(1)

where u is an algebraic number and $\alpha \in \mathbb{Z}$.

Observe that ${H}_{n}^{\left(1\right)}\left(x;u\right)={H}_{n}\left(x;u\right)$, which denotes the Frobenius-Euler polynomials and ${H}_{n}^{\left(\alpha \right)}\left(0;u\right)={H}_{n}^{\left(\alpha \right)}\left(u\right)$, which denotes the Frobenius-Euler numbers of order α. ${H}_{n}\left(x;-1\right)={E}_{n}\left(x\right)$, which denotes the Euler polynomials (cf. [124]).

Definition 1.1 (for details, see [16, 17])

Let $a,b,c\in {\mathbb{R}}^{+}$, $a\ne b$, $x\in \mathbb{R}$. The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function:

${\left(\frac{{a}^{t}-u}{\lambda {b}^{t}-u}\right)}^{\alpha }{c}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(\alpha \right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}.$
(2)

Remark 1.2 If we set $x=0$ and $\alpha =1$ in (2), we get

$\frac{{a}^{t}-u}{\lambda {b}^{t}-u}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!},$
(3)

where ${\mathcal{H}}_{n}\left(u;\lambda ;a,b,c\right)$ denotes the generalized Apostol-type Frobenius-Euler numbers (cf. [17]).

## 2 New identities

In this section, we derive many new identities related to the generalized Apostol-type Frobenius-Euler numbers and polynomials of order α.

Theorem 2.1 Let $\alpha ,\beta \in \mathbb{Z}$. Each of the following relationships holds true:

(4)
(5)
(6)

and

${\mathcal{H}}_{n}^{\left(-\alpha \right)}\left(x;{u}^{2};{a}^{2},{b}^{2},{c}^{2};{\lambda }^{2}\right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){\mathcal{H}}_{k}^{\left(-\alpha \right)}\left(x;u;a,b,c;\lambda \right){\mathcal{H}}_{n-k}^{\left(-\alpha \right)}\left(x;-u;a,b,c;\lambda \right).$
(7)

Proof of (6) From (2),

$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(-\alpha \right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(\alpha \right)}\left(y;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}={c}^{\left(x+y\right)t}.$
(8)

Therefore,

$\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){\mathcal{H}}_{n-k}^{\left(\alpha \right)}\left(y;u;a,b,c;\lambda \right){\mathcal{H}}_{k}^{\left(-\alpha \right)}\left(x;u;a,b,c;\lambda \right)\right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}{\left(xlnc\right)}^{n}\frac{{t}^{n}}{n!}.$

Thus, by using the Cauchy product in (8) and then equating the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the resulting equation, we obtain the desired result.

The proofs of (4), (5) and (7) are the same as that of (2), thus we omit them. □

Observe that in (6) we have

${\left(\left(x+y\right)lnc\right)}^{n}={\left({\mathcal{H}}^{\left(\alpha \right)}\left(y;u;a,b,c;\lambda \right)+{\mathcal{H}}^{\left(-\alpha \right)}\left(x;u;a,b,c;\lambda \right)\right)}^{n},$

where ${\left({\mathcal{H}}^{\left(\alpha \right)}\left(y;u;a,b,c;\lambda \right)\right)}^{n}$ is replaced by ${\mathcal{H}}_{n}^{\left(\alpha \right)}\left(y;u;a,b,c;\lambda \right)$.

Theorem 2.2 Let $\alpha \in \mathbb{N}$. Then we have

$\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0}{}{\alpha }{k}\right){\left(-u\right)}^{\alpha -k}{\left(xlnc+klna\right)}^{n}=\sum _{p=0}^{n}\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0}{}{n}{p}\right)\left(\genfrac{}{}{0}{}{\alpha }{k}\right){\left(-u\right)}^{\alpha -k}{\left(klnb\right)}^{p}{\mathcal{H}}_{n-p}^{\left(\alpha \right)}\left(x;u;a,b,c;\lambda \right).$

Proof By using (2), we get

$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0}{}{\alpha }{k}\right){\left(-u\right)}^{\alpha -k}{\left(xlnc+klna\right)}^{n}\right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{p=0}^{n}\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0}{}{n}{p}\right)\left(\genfrac{}{}{0}{}{\alpha }{k}\right){\left(-u\right)}^{\alpha -k}{\left(klnb\right)}^{p}{\mathcal{H}}_{n-p}^{\left(\alpha \right)}\left(x;u;a,b,c;\lambda \right)\right)\frac{{t}^{n}}{n!}.\hfill \end{array}$

By equating the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the resulting equation, we obtain the desired result. □

Theorem 2.3 The following relationship holds true:

(9)

Proof We set

$\begin{array}{c}\left(2u-1\right)\frac{{a}^{t}-u}{\lambda {b}^{t}-u}{c}^{xt}\frac{{a}^{t}-\left(1-u\right)}{\lambda {b}^{t}-\left(1-u\right)}{c}^{yt}\hfill \\ \phantom{\rule{1em}{0ex}}=\left({a}^{t}-u\right)\left({a}^{t}-\left(1-u\right)\right){c}^{\left(x+y\right)t}\left(\frac{1}{\lambda {b}^{t}-u}-\frac{1}{\lambda {b}^{t}-\left(1-u\right)}\right).\hfill \end{array}$

From the above equation, we see that

$\begin{array}{c}\left(2u-1\right)\left(\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\right)\left(\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(y;1-u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left({a}^{t}-1+u\right)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(x+y;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}-\left({a}^{t}-u\right)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(x+y;1-u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}.\hfill \end{array}$

Therefore,

$\begin{array}{c}\left(2u-1\right)\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^{n}\left(\genfrac{}{}{0}{}{n}{r}\right){\mathcal{H}}_{r}\left(x;u;a,b,c;\lambda \right){\mathcal{H}}_{n-r}\left(y;1-u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{1em}{0ex}}=\left(u-1\right)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(x+y;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}+u\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(x+y;1-u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{2em}{0ex}}+\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^{n}\left(\genfrac{}{}{0}{}{n}{r}\right){\left(lna\right)}^{n-r}{\mathcal{H}}_{r}\left(x+y;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{2em}{0ex}}-\sum _{n=0}^{\mathrm{\infty }}\sum _{r=0}^{n}\left(\genfrac{}{}{0}{}{n}{r}\right){\left(lna\right)}^{n-r}{\mathcal{H}}_{r}\left(x+y;1-u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}.\hfill \end{array}$

Comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we arrive at the desired result. □

Remark 2.4 By substituting $a=1$, $b=c=e$, $\lambda =1$ into Theorem 2.3, we get Carlitz’s results (for details, see [[1], Eq. 2.19]) as follows:

$\begin{array}{c}\left(2u-1\right)\sum _{r=0}^{n}\left(\genfrac{}{}{0}{}{n}{r}\right){H}_{r}\left(x;u\right){H}_{n-r}\left(y;1-u\right)\hfill \\ \phantom{\rule{1em}{0ex}}=\left(u-1\right){H}_{n}\left(x+y;u\right)+u{H}_{n}\left(x+y;1-u\right)+{H}_{n}\left(x+y;u\right)-{H}_{n}\left(x+y;1-u\right).\hfill \end{array}$

We give the following generating function of the polynomials ${Y}_{n}\left(x;\lambda ;a\right)$:

$\frac{t}{\lambda {a}^{t}-1}{a}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{Y}_{n}\left(x;\lambda ;a\right)\frac{{t}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(a\ge 1\right)$
(10)

(cf. [16, 17]). We also note that

${Y}_{n}\left(0;\lambda ;a\right)={Y}_{n}\left(\lambda ;a\right).$

If we substitute $x=0$ and $a=1$ into (10), we see that

${Y}_{n}\left(\lambda ;1\right)=\frac{1}{\lambda -1}.$

Theorem 2.5 The generalized Apostol-type Frobenius-Euler polynomial holds true as follows:

(11)

Proof Substituting $c=b$ for $\alpha =1$ into (2) and taking derivative with respect to t, we obtain

$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n+1}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{a}^{t}lna}{{a}^{t}-u}\frac{{a}^{t}-u}{\lambda {b}^{t}-u}{b}^{xt}+\frac{lnb\lambda {b}^{t}}{{a}^{t}-u}{\left(\frac{{a}^{t}-u}{\lambda {b}^{t}-u}\right)}^{2}{b}^{xt}+ln\left({b}^{x}\right)\frac{{a}^{t}-u}{\lambda {b}^{t}-u}{b}^{xt}.\hfill \end{array}$

Using (10), we have

$\begin{array}{rcl}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n+1}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n}}{n!}& =& \frac{ln\left({a}^{\frac{1}{u}}\right)}{t}\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){Y}_{n-k}\left(1;\frac{1}{u};a\right){\mathcal{H}}_{k}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n}}{n!}\\ +\frac{ln\left({b}^{\frac{\lambda }{u}}\right)}{t}\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){Y}_{n-k}\left(\frac{1}{u};a\right){\mathcal{H}}_{k}^{\left(2\right)}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n}}{n!}\\ +ln\left({b}^{x}\right)\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n}}{n!}.\end{array}$

Thus, after some elementary calculations, we arrive at (11). □

Theorem 2.6 Let $|u|<1$ and $m\in \mathbb{N}$. Then we have

${\mathcal{H}}^{\left(-m\right)}\left(u;a,b,c;\lambda \right)=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){\mathcal{H}}_{k}^{\left(-\alpha \right)}\left(-x;u;a,b,c;\lambda \right){\mathcal{H}}_{n-k}^{\left(\alpha -m\right)}\left(x;u;a,b,c;\lambda \right).$
(12)

Proof In (2), we replace α by −α, then we set

${\left(\frac{{a}^{t}-u}{\lambda {b}^{t}-u}\right)}^{-\alpha }{c}^{\left(-x\right)t}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(\alpha -m\right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}={\left(\frac{{a}^{t}-u}{\lambda {b}^{t}-u}\right)}^{-m}.$

By using (2), we get

$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(-\alpha \right)}\left(-x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(\alpha -m\right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(-m\right)}\left(u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}.$

Therefore,

$\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){\mathcal{H}}_{k}^{\left(-\alpha \right)}\left(-x;u;a,b,c;\lambda \right){\mathcal{H}}_{n-k}^{\left(\alpha -m\right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(-m\right)}\left(u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}.$

Comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we arrive at (12). □

## 3 Interpolation function

In this section, we give a recurrence relation between the generalized Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function. Recently, many authors have studied not only the Hurwitz-Lerch zeta function, but also its generalizations, for example (among others), Srivastava [19], Srivastava and Choi [24] and also Garg et al. [6]. The generalization of the Hurwitz-Lerch zeta function $\mathrm{\Phi }\left(z,s,a\right)$ is given as follows:

${\mathrm{\Phi }}_{\mu ,\nu }^{\left(\rho ,\sigma \right)}\left(z,s,a\right):=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\mu \right)}_{\rho n}}{{\left(\nu \right)}_{\sigma n}}\frac{{z}^{n}}{{\left(n+a\right)}^{s}}$

($\mu \in \mathbb{C}$, $a,\upsilon \in \mathbb{C}\mathrm{\setminus }{\mathbb{Z}}_{0}^{-}$, $\rho ,\sigma \in {\mathbb{R}}^{+}$, $\rho <\sigma$ when $s,z\in \mathbb{C}$ ($|z|<1$); $\rho =\sigma$ and $\mathrm{\Re }\left(s-\mu +\nu \right)>0$ when $|z|=1$). It is obvious that

${\mathrm{\Phi }}_{\mu ,1}^{\left(1,1\right)}\left(z,s,a\right)={\mathrm{\Phi }}_{\mu }^{\ast }\left(z,s,a\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\mu \right)}_{n}}{n!}\frac{{z}^{n}}{{\left(n+a\right)}^{s}}$
(13)

and

${\mathrm{\Phi }}_{n}^{\ast }\left(z,s,a\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(n\right)}_{n}}{n!}\frac{{z}^{n}}{{\left(n+a\right)}^{s}}=\mathrm{\Phi }\left(z,s,a\right),$

where $\mathrm{\Phi }\left(z,s,a\right)$ denotes the Lerch-Zeta function (cf. [6, 19, 21, 24]).

Relation between the generalized Apostol-type Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function is given as follows.

Theorem 3.1 Let $|\frac{\lambda }{u}|<1$. We have

${\mathcal{H}}_{n}^{\left(\alpha \right)}\left(x;u;a,b,c;\lambda \right)=\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0}{}{\alpha }{k}\right){\left(-u\right)}^{\alpha -k-1}\mathfrak{G}\left(-n;x,\frac{\lambda }{u};a,b,c;\alpha ,k\right),$
(14)

where

$\mathfrak{G}\left(s;x,\beta ;a,b,c;\alpha ,j\right)=\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m}\right)\frac{{\beta }^{m}}{{\left(xlnc+jlna+mlnb\right)}^{s}},\phantom{\rule{1em}{0ex}}|\beta |<1.$

Proof From (2), we have

$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(\alpha \right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}=\sum _{j=0}^{\alpha }\left(\genfrac{}{}{0}{}{\alpha }{j}\right){\left(-u\right)}^{\alpha -j-1}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m}\right){\left(\frac{\lambda }{u}\right)}^{m}{e}^{\alpha \left(xlnc+klna+mlnb\right)}.$

Therefore,

$\begin{array}{c}\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(\alpha \right)}\left(x;u;a,b,c;\lambda \right)\frac{{t}^{n}}{n!}\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{n=0}^{\mathrm{\infty }}\sum _{k=0}^{\alpha }\left(\genfrac{}{}{0}{}{\alpha }{k}\right){\left(-u\right)}^{\alpha -k-1}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{m+\alpha -1}{m}\right){\left(\frac{\lambda }{u}\right)}^{m}{\left(xlnc+klna+mlnb\right)}^{n}\frac{{t}^{n}}{n!}.\hfill \end{array}$

Comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we have arrive at (14). □

Remark 3.2 By substituting $a=1$, $b=c=e$ into (14), we have

${\mathcal{H}}_{n}^{\left(\alpha \right)}\left(x;u;\lambda \right)=-\frac{{\left(1-u\right)}^{\alpha }}{u}\mathfrak{G}\left(-n;x,\frac{\lambda }{u};1,e,e;\alpha ,1\right)=-\frac{{\left(1-u\right)}^{\alpha }}{u}\mathrm{\Phi }\left(\frac{\lambda }{u},-n,x\right),$

where

$\mathfrak{G}\left(-n;x,\frac{\lambda }{u};1,e,e;\alpha ,1\right)=\mathrm{\Phi }\left(\frac{\lambda }{u},-n,x\right).$

Remark 3.3 The function $\mathfrak{G}\left(s;x,\beta ;a,b,c;\alpha ,j\right)$ is an interpolation function of the generalized Apostol-type Frobenius-Euler polynomials of order α at negative integers, which is given by the analytic continuation of the $\mathfrak{G}\left(s;x,\beta ;a,b,c;\alpha ,j\right)$ for $s=-n$, $n\in \mathbb{N}$.

## 4 Relations between Array-type polynomials, Apostol-Bernoulli polynomials and generalized Apostol-type Frobenius-Euler polynomial

In [17], Simsek constructed the generalized λ-Stirling type numbers of the second kind $\mathcal{S}\left(n,v;a,b;\lambda \right)$ by means of the following generating function:

${f}_{S,v}\left(t;a,b;\lambda \right)=\frac{{\left(\lambda {b}^{t}-{a}^{t}\right)}^{v}}{v!}=\sum _{n=0}^{\mathrm{\infty }}\mathcal{S}\left(n,v;a,b;\lambda \right)\frac{{t}^{n}}{n!}.$
(15)

The generating function for these polynomials ${\mathcal{S}}_{v}^{n}\left(x;a,b;\lambda \right)$ is given by

${g}_{v}\left(x,t;a,b;\lambda \right)=\frac{1}{v!}{\left(\lambda {b}^{t}-{a}^{t}\right)}^{v}{b}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{S}}_{v}^{n}\left(x;a,b;\lambda \right)\frac{{t}^{n}}{n!}$
(16)

(cf. [17]).

The generalized Apostol-Bernoulli polynomials were defined by Srivastava et al. [[22], p.254, Eq. (20)] as follows.

Let $a,b,c\in {\mathbb{R}}^{+}$ with $a\ne b$, $x\in \mathbb{R}$ and $n\in {\mathbb{N}}_{0}$. Then the generalized Bernoulli polynomials ${\mathfrak{B}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;a,b,c\right)$ of order $\alpha \in \mathbb{Z}$ are defined by means of the following generating functions:

${f}_{B}\left(x,a,b,c;\lambda ;\alpha \right)={\left(\frac{t}{\lambda {b}^{t}-{a}^{t}}\right)}^{\alpha }{c}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathfrak{B}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;a,b,c\right)\frac{{t}^{n}}{n!},$
(17)

where

$|tln\left(\frac{a}{b}\right)+ln\lambda |<2\pi .$

We note that ${\mathfrak{B}}_{n}^{\left(1\right)}\left(x;\lambda ;a,b,c\right)={\mathfrak{B}}_{n}\left(x;\lambda ;a,b,c\right)$ and also ${\mathfrak{B}}_{n}\left(x;\lambda ;1,e,e\right)={B}_{n}\left(x;\lambda \right)$, which denotes the Apostol-Bernoulli polynomials (cf. [124]).

Theorem 4.1 Let v be an integer. Then we have

${\mathcal{H}}_{n-v}^{\left(-\nu \right)}\left(x;u;a,b,c;\lambda \right)=\frac{\nu !}{{u}^{2\nu }{\left(n\right)}_{v}}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){\mathcal{S}}_{v}^{n}\left(x,1,b;\frac{\lambda }{u}\right){Y}_{n-k}^{\left(\nu \right)}\left(\frac{1}{u};a\right).$

Proof Replacing c by b in (2) and after some calculations, we have

$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n}^{\left(-v\right)}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n+v}}{n!}=\frac{\nu !}{{u}^{2\nu }}\sum _{n=0}^{\mathrm{\infty }}{S}_{\nu }^{n}\left(x,1,b;\frac{\lambda }{u}\right)\frac{{t}^{n}}{n!}\sum _{n=0}^{\mathrm{\infty }}{Y}_{n}^{\left(\nu \right)}\left(\frac{1}{u};a\right)\frac{{t}^{n}}{n!}.$

Comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we arrive at the desired result. □

Corollary 4.2

${\mathcal{H}}_{n-v}^{\left(-\nu \right)}\left(x;u;a,b,c;\lambda \right)=\frac{\nu !}{{u}^{2\nu }{\left(n\right)}_{\alpha }}\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right)\mathcal{S}\left(k,\nu ,1,b;\frac{\lambda }{u}\right){\mathfrak{B}}_{n-k}\left(x,a,b;\frac{\lambda }{u}\right).$

Proof Replacing c by b in (2) and after some calculations, we have

$\sum _{n=0}^{\mathrm{\infty }}{\mathcal{H}}_{n-v}^{\left(-v\right)}\left(x;u;a,b,b;\lambda \right)\frac{{t}^{n+v}}{n!}=\frac{\nu !}{{u}^{2\nu }}\sum _{n=0}^{\mathrm{\infty }}\mathcal{S}\left(n,\nu ,1,b;\frac{\lambda }{u}\right)\frac{{t}^{n}}{n!}\sum _{n=0}^{\mathrm{\infty }}{\mathfrak{B}}_{n}\left(x,a,b;\frac{\lambda }{u}\right)\frac{{t}^{n}}{n!}.$

Comparing the coefficients of $\frac{{t}^{n}}{n!}$ on both sides of the above equation, we arrive at the desired result. □

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## Acknowledgements

Dedicated to Professor Hari M. Srivastava.

All authors are partially supported by Research Project Offices Akdeniz Universities.

## Author information

Authors

### Corresponding author

Correspondence to Yilmaz Simsek.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

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Reprints and Permissions

Kurt, B., Simsek, Y. On the generalized Apostol-type Frobenius-Euler polynomials. Adv Differ Equ 2013, 1 (2013). https://doi.org/10.1186/1687-1847-2013-1

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1847-2013-1

### Keywords

• Frobenius-Euler polynomials
• Hermite-based Frobenius-Euler polynomials
• Hermite-based Apostol-Euler polynomials
• Apostol-Euler polynomials
• Hurwitz-Lerch zeta function