Theory and Modern Applications

# Some results for Apostol-type polynomials associated with umbral algebra

## Abstract

A family of the Apostol-type polynomials was introduced and investigated recently by Luo and Srivastava (see (Appl. Math. Comput. 217:5702-5728, 2011)). In this paper, we study this polynomial family on P, the algebra of polynomials in a single variable x over all linear functional on P. By using the way of the umbral algebra, we obtain some fundamental properties of the generalized Apostol-type polynomials. We also show some special cases which include the corresponding results of Dere and Simsek etc.

MSC:05A40, 11B68, 05A10, 05A15.

## 1 Introduction, definitions and motivation

Throughout this paper, we make use of the following conventional notations: $\mathbb{N}=\left\{1,2,3,\dots \right\}$ denotes the set of natural numbers, denotes the set of complex numbers.

The classical Bernoulli polynomials ${B}_{n}\left(x\right)$, the classical Euler polynomials ${E}_{n}\left(x\right)$ and the classical Genocchi polynomials ${G}_{n}\left(x\right)$, together with their familiar generalizations ${B}_{n}^{\left(\alpha \right)}\left(x\right)$, ${E}_{n}^{\left(\alpha \right)}\left(x\right)$ and ${G}_{n}^{\left(\alpha \right)}\left(x\right)$ of order α, are usually defined by means of the following generating functions (see, for details, [, pp.532-533] and ):

${\left(\frac{z}{{e}^{z}-1}\right)}^{\alpha }{e}^{xz}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n}^{\left(\alpha \right)}\left(x\right)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(|z|<2\pi \right),$
(1.1)
${\left(\frac{2}{{e}^{z}+1}\right)}^{\alpha }{e}^{xz}=\sum _{n=0}^{\mathrm{\infty }}{E}_{n}^{\left(\alpha \right)}\left(x\right)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(|z|<\pi \right)$
(1.2)

and

${\left(\frac{2z}{{e}^{z}+1}\right)}^{\alpha }{e}^{xz}=\sum _{n=0}^{\mathrm{\infty }}{G}_{n}^{\left(\alpha \right)}\left(x\right)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(|z|<\pi \right).$
(1.3)

It is easy to see that ${B}_{n}\left(x\right)$, ${E}_{n}\left(x\right)$ and ${G}_{n}\left(x\right)$ are given, respectively, by

$\begin{array}{r}{B}_{n}\left(x\right):={B}_{n}^{\left(1\right)}\left(x\right),\phantom{\rule{2em}{0ex}}{E}_{n}\left(x\right):={E}_{n}^{\left(1\right)}\left(x\right)\phantom{\rule{1em}{0ex}}\text{and}\\ {G}_{n}\left(x\right):={G}_{n}^{\left(1\right)}\left(x\right)\phantom{\rule{1em}{0ex}}\left(n\in {\mathbb{N}}_{0}:=\mathbb{N}\cup \left\{0\right\}\right).\end{array}$
(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

${B}_{n}:={B}_{n}\left(0\right)={B}_{n}^{\left(1\right)}\left(0\right),\phantom{\rule{2em}{0ex}}{E}_{n}:={E}_{n}\left(0\right)={E}_{n}^{\left(1\right)}\left(0\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{G}_{n}:={G}_{n}\left(0\right)={G}_{n}^{\left(1\right)}\left(0\right),$
(1.5)

respectively.

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

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

(1.6)

with, of course,

${B}_{n}\left(x\right)={\mathcal{B}}_{n}\left(x;1\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{B}}_{n}\left(\lambda \right):={\mathcal{B}}_{n}\left(0;\lambda \right),$
(1.7)

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

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

Definition 1.2 (Luo and Srivastava )

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

(1.8)

with, of course,

${B}_{n}^{\left(\alpha \right)}\left(x\right)={\mathcal{B}}_{n}^{\left(\alpha \right)}\left(x;1\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{B}}_{n}^{\left(\alpha \right)}\left(\lambda \right):={\mathcal{B}}_{n}^{\left(\alpha \right)}\left(0;\lambda \right),$
(1.9)

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

In this sequel, Luo  gave an analogous extension of the generalized Euler polynomials which is the so-called Apostol-Euler polynomials of order α.

Definition 1.3 (Luo )

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

${\left(\frac{2}{\lambda {e}^{z}+1}\right)}^{\alpha }\cdot {e}^{xz}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(|z|<|log\left(-\lambda \right)|\right)$
(1.10)

with, of course,

${E}_{n}^{\left(\alpha \right)}\left(x\right)={\mathcal{E}}_{n}^{\left(\alpha \right)}\left(x;1\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{E}}_{n}^{\left(\alpha \right)}\left(\lambda \right):={\mathcal{E}}_{n}^{\left(\alpha \right)}\left(0;\lambda \right),$
(1.11)

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

On the subject of the Genocchi polynomials ${G}_{n}\left(x\right)$ and their various extensions, a remarkably large number of investigations have appeared in the literature (see, for example, ). Moreover, Luo (see ) 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 ${\mathcal{G}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)$ ($\lambda \in \mathbb{C}$) of order α ($\alpha \in \mathbb{N}$) are defined by means of the following generating function:

${\left(\frac{2z}{\lambda {e}^{z}+1}\right)}^{\alpha }\cdot {e}^{xz}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(|z|<|log\left(-\lambda \right)|\right)$
(1.12)

with, of course,

$\begin{array}{r}{G}_{n}^{\left(\alpha \right)}\left(x\right)={\mathcal{G}}_{n}^{\left(\alpha \right)}\left(x;1\right),\phantom{\rule{2em}{0ex}}{\mathcal{G}}_{n}^{\left(\alpha \right)}\left(\lambda \right):={\mathcal{G}}_{n}^{\left(\alpha \right)}\left(0;\lambda \right),\\ {\mathcal{G}}_{n}\left(x;\lambda \right):={\mathcal{G}}_{n}^{\left(1\right)}\left(x;\lambda \right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathcal{G}}_{n}\left(\lambda \right):={\mathcal{G}}_{n}^{\left(1\right)}\left(\lambda \right),\end{array}$
(1.13)

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

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

(1.14)

It is found from  that Ozden further gave an extension of the above definition (1.14) as follows:

Definition 1.5

(1.15)

The author  obtained a unified relation between the ${\mathcal{Y}}_{n,\beta }^{\left(\alpha \right)}\left(x;\kappa ,a,b\right)$ and the Gauss hypergeometric function ${}_{2}F_{1}\left(a,b;c;z\right)$, and gave some identities of ${\mathcal{Y}}_{n,\beta }^{\left(\alpha \right)}\left(x;\kappa ,a,b\right)$.

Recently, Luo and Srivastava  introduced more general unification (and generalization) of the above-mentioned three families of the generalized Apostol-type polynomials.

Definition 1.6 (Luo and Srivastava )

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

${\left(\frac{{2}^{\mu }{z}^{\nu }}{\lambda {e}^{z}+1}\right)}^{\alpha }{e}^{xz}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)\frac{{z}^{n}}{n!}\phantom{\rule{1em}{0ex}}\left(|z|<|log\left(-\lambda \right)|\right).$
(1.16)

Clearly, we have

${\mathcal{B}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)={\left(-1\right)}^{\alpha }{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;-\lambda ;0;1\right)\phantom{\rule{1em}{0ex}}\left(\alpha \in \mathbb{N}\right),$
(1.17)
${\mathcal{E}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)={\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;1;0\right)\phantom{\rule{1em}{0ex}}\left(\alpha \in \mathbb{C}\right),$
(1.18)
${\mathcal{G}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)={\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;1;1\right)\phantom{\rule{1em}{0ex}}\left(\alpha \in \mathbb{N}\right),$
(1.19)
${\mathcal{Y}}_{n,\beta }\left(x;\kappa ,a,b\right)=-\frac{1}{{a}^{b}}{\mathcal{F}}_{n}^{\left(1\right)}\left(x;-{\left(\frac{\beta }{a}\right)}^{b};1-\kappa ;\kappa \right)$
(1.20)

and

${\mathcal{Y}}_{n,\beta }^{\left(\alpha \right)}\left(x;\kappa ,a,b\right)={\left(-1\right)}^{\alpha }\frac{1}{{a}^{b\alpha }}{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;-{\left(\frac{\beta }{a}\right)}^{b};1-\kappa ;\kappa \right).$
(1.21)

In [5, 6, 17, 18], the authors have researched some elementary properties of the Apostol-type polynomials, and some relationships among the Apostol-type polynomials. More investigations about this subject can be found in [13, 15, 16, 1930].

The aim of this paper is to study the generalized Apostol-type polynomials ${\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)$ on the umbral algebra by using the way as the reference . We research some fundamental properties of this polynomial family. Some special cases, which include the corresponding results , are also considered.

## 2 Umbral algebra of Roman

We can use the following notations and definitions, which are given by Roman , pp.1-125].

Let P be the algebra of polynomials in a single variable x over the field of complex numbers. Let ${P}^{\ast }$ be the vector space of all linear functionals on P. Let $〈L|p\left(x\right)〉$ be the action of a linear functional L on a polynomial $p\left(x\right)$. Let denote the algebra of formal power series

$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{a}_{k}}{k!}{t}^{k}.$
(2.1)

Such algebra is called umbral algebra. Each $f\in \mathcal{F}$ defines a linear functional on P and

${a}_{k}=〈f\left(t\right)|{t}^{k}〉$
(2.2)

for all $k\ge 0$.

The order $o\left(f\left(t\right)\right)$ of a power series $f\left(t\right)$ is the smallest integer k for which the coefficient of ${t}^{k}$ does not vanish. A series $f\left(t\right)$ for which $o\left(f\left(t\right)\right)=1$ will be called a delta series. When we are considering a delta series $f\left(t\right)$ in as a linear functional, we will refer to it as a delta functional.

It is well known that $〈{t}^{k}|{x}^{n}〉=n!{\delta }_{n,k}$, where ${\delta }_{n,k}$ denotes the Kronecker symbol. For all $f\left(t\right)$ in ,

$f\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈f\left(t\right)|{x}^{k}〉}{k!}{t}^{k}.$

Let $f\left(t\right)$ and $g\left(t\right)$ be in . Then we have

$〈f\left(t\right)g\left(t\right)|p\left(x\right)〉=〈f\left(t\right)|g\left(t\right)p\left(x\right)〉.$
(2.3)

For $y\in \mathbb{C}$, then the evaluation functional is defined to be the power series ${e}^{yt}$. By (2.2), we have

$〈{e}^{yt}|p\left(x\right)〉=p\left(y\right)$
(2.4)

for all $p\left(x\right)$ in P. The forward difference functional is the delta functional ${e}^{yt}-1$ and

$〈{e}^{yt}-1|p\left(x\right)〉=p\left(y\right)-p\left(0\right).$
(2.5)

The Abel functional is the delta functional $t{e}^{yt}$. We have

$〈t{e}^{yt}|p\left(x\right)〉={p}^{\prime }\left(y\right).$

The Sheffer polynomials are defined by means of the following generating function

$\sum _{k=0}^{\mathrm{\infty }}\frac{{s}_{k}\left(x\right)}{k!}{t}^{k}=\frac{1}{g\left(t\right)}{e}^{xt}.$
(2.6)

Roman  proved the following theorem which is represented by the Sheffer polynomials (or Sheffer sequences) explicitly.

Theorem 2.1 Let $f\left(t\right)$ be a delta series and let $g\left(t\right)$ be an invertible series. Then there exists a unique sequence ${s}_{n}\left(x\right)$ of polynomials satisfying the orthogonality conditions

$〈g\left(t\right)f{\left(t\right)}^{k}|{s}_{n}\left(x\right)〉=n!{\delta }_{n,k}$
(2.7)

for all $k\in {\mathbb{N}}_{0}$.

The sequence ${s}_{n}\left(x\right)$ in (2.7) is the Sheffer polynomials for pair $\left(g\left(t\right),f\left(t\right)\right)$, where $g\left(t\right)$ must be invertible and $f\left(t\right)$ must be delta series. The Sheffer polynomials for pair $\left(g\left(t\right),t\right)$ is the Appell polynomials or the Appell sequences for $g\left(t\right)$.

The Appell polynomials, the Bernoulli polynomials, the Euler polynomials, the Genocchi polynomials and the Genocchi polynomials of higher order belong to the family of the Sheffer polynomials (cf. [31, 3436]).

The Sheffer polynomials satisfy the following relations:

${s}_{n}\left(x\right)=g{\left(t\right)}^{-1}{x}^{n},$
(2.8)

derivative formula

$t{s}_{n}\left(x\right)={s}_{n}^{\prime }\left(x\right)=n{s}_{n-1}\left(x\right),$
(2.9)

recurrence formula

${s}_{n+1}\left(x\right)=\left(x-\frac{{g}^{\prime }\left(t\right)}{g\left(t\right)}\right){s}_{n}\left(x\right),$
(2.10)

expansion theorem

$h\left(t\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{〈h\left(t\right)|{s}_{k}\left(x\right)〉}{k!}g\left(t\right){t}^{k},$
(2.11)

multiplication theorem, for $\alpha \ne 0$,

${s}_{n}\left(\alpha x\right)={\alpha }^{n}\frac{g\left(t\right)}{g\left(\frac{t}{\alpha }\right)}{s}_{n}\left(x\right),$
(2.12)

and

$〈h\left(t\right)|p\left(ax\right)〉=〈h\left(at\right)|p\left(x\right)〉.$
(2.13)

## 3 The Apostol-type polynomials on ℱ

We see from Definition 1.6 and (2.6) that the generalized Apostol-type polynomials ${\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)$ also belong to the Sheffer polynomials where $g\left(t\right)={\left(\frac{\lambda {e}^{t}+1}{{2}^{\mu }{t}^{\nu }}\right)}^{\alpha }$.

In this section, by using the properties of the Sheffer sequences and also the Appell sequences, we prove many fundamental properties of the generalized Apostol-type polynomials ${\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)$ defined by (1.16).

By using (2.8) and (1.16), we arrive at the following lemma.

Lemma 3.1

${\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)={\left(\frac{{2}^{\mu }{t}^{\nu }}{\lambda {e}^{t}+1}\right)}^{\alpha }{x}^{n}.$
(3.1)

Theorem 3.2

$〈\frac{{\left(\lambda {e}^{t}+1\right)}^{k}}{{t}^{\nu -1}}|{\mathcal{F}}_{n}^{\left(1\right)}\left(x;\lambda ;\mu ;\nu \right)〉={2}^{\mu }{\lambda }^{k-1}n\left(k-1\right)!\sum _{j=0}^{k-1}{\left(1+\frac{1}{\lambda }\right)}^{k-j-1}\frac{S\left(n-1,j\right)}{\left(k-j-1\right)!},$
(3.2)

where ${\mathcal{F}}_{n}^{\left(1\right)}\left(x;\lambda ;\mu ;\nu \right)$ and $S\left(a,b\right)$ denote the first-order generalized Apostol-type polynomials and the Stirling numbers of the second kind, respectively.

Proof By Lemma 3.1, we obtain

$〈\frac{{\left(\lambda {e}^{t}+1\right)}^{k}}{{t}^{\nu -1}}|{\mathcal{F}}_{n}^{\left(1\right)}\left(x;\lambda ;\mu ;\nu \right)〉=〈\frac{{\left(\lambda {e}^{t}+1\right)}^{k}}{{t}^{\nu -1}}|\frac{{2}^{\mu }{t}^{\nu }}{\lambda {e}^{t}+1}{x}^{n}〉.$

By using (2.3) and (2.9), we get

$\begin{array}{r}〈\frac{{\left(\lambda {e}^{t}+1\right)}^{k}}{{t}^{\nu -1}}|{\mathcal{F}}_{n}^{\left(1\right)}\left(x;\lambda ;\mu ;\nu \right)〉\\ \phantom{\rule{1em}{0ex}}={2}^{\mu }{\lambda }^{k-1}n\sum _{j=0}^{k-1}\frac{\left(k-1\right)!}{\left(k-j-1\right)!}{\left(1+\frac{1}{\lambda }\right)}^{k-j-1}〈\frac{{\left({e}^{t}-1\right)}^{j}}{j!}|{x}^{n-1}〉.\end{array}$
(3.3)

Setting

$S\left(n-1,j\right)=\frac{1}{j!}〈{\left({e}^{t}-1\right)}^{j}|{x}^{n-1}〉,$

where $S\left(n-1,j\right)$ denotes the Stirling numbers of second kind (cf. [, p.59]) in (3.3), we arrive at the desired result. □

We deduce the following formulas.

Letting $\lambda ↦-\lambda$, taking $\mu =0$ and $\nu =1$ in (3.2) and noting relation (1.17), we deduce the following result.

Corollary 3.3 (see [, Remark 19])

$〈{\left(1-\lambda {e}^{t}\right)}^{k}|{\mathcal{B}}_{n}\left(x;\lambda \right)〉={\left(-1\right)}^{k}{\lambda }^{k-1}n\left(k-1\right)!\sum _{j=0}^{k-1}{\left(1-\frac{1}{\lambda }\right)}^{k-j-1}\frac{S\left(n-1,j\right)}{\left(k-j-1\right)!},$
(3.4)

where ${\mathcal{B}}_{n}\left(x;\lambda \right)$ and $S\left(a,b\right)$ denote the Apostol-Bernoulli polynomials and the Stirling numbers of the second kind, respectively.

Taking $\mu =1$ and $\nu =0$ in (3.2) and noting relation (1.18), we deduce the following result.

Corollary 3.4 (see [, Remark 21])

$〈t{\left(\lambda {e}^{t}+1\right)}^{k}|{\mathcal{E}}_{n}\left(x;\lambda \right)〉=2{\lambda }^{k-1}n\left(k-1\right)!\sum _{j=0}^{k-1}{\left(1+\frac{1}{\lambda }\right)}^{k-j-1}\frac{S\left(n-1,j\right)}{\left(k-j-1\right)!},$
(3.5)

where ${\mathcal{E}}_{n}\left(x;\lambda \right)$ and $S\left(a,b\right)$ denote the Apostol-Euler polynomials and the Stirling numbers of the second kind, respectively.

Taking $\mu =\nu =1$ in (3.2) and noting relation (1.19), we deduce the following result.

Corollary 3.5 (see [, Remark 20])

$〈{\left(\lambda {e}^{t}+1\right)}^{k}|{\mathcal{G}}_{n}\left(x;\lambda \right)〉=2{\lambda }^{k-1}n\left(k-1\right)!\sum _{j=0}^{k-1}{\left(1+\frac{1}{\lambda }\right)}^{k-j-1}\frac{S\left(n-1,j\right)}{\left(k-j-1\right)!},$
(3.6)

where ${\mathcal{G}}_{n}\left(x;\lambda \right)$ and $S\left(a,b\right)$ denote the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, respectively.

Setting $\lambda =1$ in (3.6), we deduce Theorem 2 in the work [, p.758, Theorem 2].

Corollary 3.6

$〈{\left({e}^{t}+1\right)}^{k}|{G}_{n}\left(x\right)〉=2n\left(k-1\right)!\sum _{j=0}^{k-1}{2}^{k-j-1}\frac{S\left(n-1,j\right)}{\left(k-j-1\right)!},$
(3.7)

where ${G}_{n}\left(x\right)$ and $S\left(a,b\right)$ denote the Genocchi polynomials and the Stirling numbers of the second kind, respectively.

Letting $k↦m$, taking $\lambda =-{\left(\frac{\beta }{a}\right)}^{b}$, $\mu =1-\kappa$, $\nu =\kappa$ in (3.2) and noting relation (1.20), thus we deduce the following formulas of the polynomials ${\mathcal{Y}}_{n,\beta }\left(x;\kappa ,a,b\right)$.

Corollary 3.7

$\begin{array}{r}〈{\left[1-{\left(\frac{\beta }{a}\right)}^{b}{e}^{t}\right]}^{m}{t}^{1-\kappa }|{\mathcal{Y}}_{n,\beta }\left(x;\kappa ,a,b\right)〉\\ \phantom{\rule{1em}{0ex}}={\left(-1\right)}^{m}{2}^{1-\kappa }{\beta }^{b\left(m-1\right)}{a}^{-bm}n\left(m-1\right)!\sum _{j=0}^{m-1}{\left[1-{\left(\frac{a}{\beta }\right)}^{b}\right]}^{m-j-1}\frac{S\left(n-1,j\right)}{\left(m-j-1\right)!},\end{array}$
(3.8)

where ${\mathcal{Y}}_{n,\beta }\left(x;\kappa ,a,b\right)$ and $S\left(a,b\right)$ denote the generalization of Apostol type polynomials defined by (1.14) and the Stirling numbers of the second kind, respectively.

By using (2.9), we arrive at the following lemma.

Lemma 3.8

$t{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)=n{\mathcal{F}}_{n-1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right).$
(3.9)

Remark 3.9 An alternative proof of Lemma 3.8 is also obtained from (1.16) by using derivative with respect to x. By Lemma 3.8, one can see that

$\frac{1}{t}{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)=\frac{1}{n+1}{\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right).$
(3.10)

Theorem 3.10

$\left(\frac{{t}^{\nu -1}}{\lambda {e}^{t}+1}\right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)=\frac{1}{{2}^{\mu }\left(n+1\right)}{\mathcal{F}}_{n+1}^{\left(\alpha +1\right)}\left(x;\lambda ;\mu ;\nu \right).$
(3.11)

Proof By Lemma 3.1, we obtain

$\left(\frac{{t}^{\nu -1}}{\lambda {e}^{t}+1}\right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)=\frac{{t}^{\nu -1}}{\lambda {e}^{t}+1}{\left(\frac{{2}^{\mu }{t}^{\nu }}{\lambda {e}^{t}+1}\right)}^{\alpha }{x}^{n}.$
(3.12)

After some calculations in the above equation, we have

$\left(\frac{{t}^{\nu -1}}{\lambda {e}^{t}+1}\right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)=\frac{1}{{2}^{\mu }t}{\left(\frac{{2}^{\mu }{t}^{\nu }}{\lambda {e}^{t}+1}\right)}^{\alpha +1}{x}^{n}.$
(3.13)

Using (1.16) and (3.10), we obtain the desired result. □

Letting $\lambda ↦-\lambda$, taking $\mu =0$ and $\nu =1$ in (3.11) and noting relation (1.17), we deduce the following result.

Corollary 3.11 (see [, Remark 32])

$\left(\frac{1}{1-\lambda {e}^{t}}\right){\mathcal{B}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)=\frac{1}{n+1}{\mathcal{B}}_{n+1}^{\left(\alpha +1\right)}\left(x;\lambda \right).$
(3.14)

Taking $\mu =1$ and $\nu =0$ in (3.11) and noting relation (1.18), we deduce the following result.

Corollary 3.12 (see [, Remark 33])

$\frac{1}{t\left(\lambda {e}^{t}+1\right)}{\mathcal{E}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)=\frac{1}{2\left(n+1\right)}{\mathcal{E}}_{n+1}^{\left(\alpha +1\right)}\left(x;\lambda \right).$
(3.15)

Taking $\mu =\nu =1$ in (3.11) and noting relation (1.19), we deduce the following result.

Corollary 3.13 (see [, Remark 34])

$\left(\frac{1}{\lambda {e}^{t}+1}\right){\mathcal{G}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)=\frac{1}{2\left(n+1\right)}{\mathcal{G}}_{n+1}^{\left(\alpha +1\right)}\left(x;\lambda \right).$
(3.16)

Setting $\lambda =1$ in the above equation, we deduce Lemma 3 in [, p.758].

Corollary 3.14

$\left(\frac{1}{{e}^{t}+1}\right){G}_{n}^{\left(\alpha \right)}\left(x\right)=\frac{1}{2\left(n+1\right)}{G}_{n+1}^{\left(\alpha +1\right)}\left(x\right).$
(3.17)

Taking $\lambda =-{\left(\frac{\beta }{a}\right)}^{b}$, $\mu =1-\kappa$, $\nu =\kappa$ in (3.11) and noting relation (1.21), we deduce

Corollary 3.15

$\frac{-{t}^{\kappa -1}}{{a}^{b}\left[1-{\left(\frac{\beta }{a}\right)}^{b}{e}^{t}\right]}{\mathcal{Y}}_{n,\beta }^{\left(\alpha \right)}\left(x;\kappa ,a,b\right)=\frac{1}{{2}^{1-\kappa }\left(n+1\right)}{\mathcal{Y}}_{n+1,\beta }^{\left(\alpha +1\right)}\left(x;\kappa ,a,b\right).$
(3.18)

An integral representation of $〈\frac{{e}^{ta}-1}{2t}|{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)〉$ is given by the following theorem.

Theorem 3.16

$〈\frac{{e}^{ta}-1}{2t}|{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)〉=\frac{1}{2}{\int }_{0}^{a}{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)\phantom{\rule{0.2em}{0ex}}dx.$
(3.19)

Proof By using Lemma 3.8, we have

$〈\frac{{e}^{ta}-1}{2t}|{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)〉=〈\frac{{e}^{ta}-1}{2t}|\frac{1}{n+1}t{\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)〉.$

By (2.3), we obtain

$〈\frac{{e}^{ta}-1}{2t}|{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)〉=\frac{1}{2\left(n+1\right)}〈{e}^{ta}-1|{\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)〉.$

Using (2.5), we obtain the desired result. □

Setting $\lambda =\mu =\nu =1$ in (3.19) and noting relation (1.19), we deduce the Theorem 3 in [, p.758].

Corollary 3.17

$〈\frac{{e}^{ta}-1}{2t}|{G}_{n}^{\left(\alpha \right)}\left(x\right)〉=\frac{1}{2}{\int }_{0}^{a}{G}_{n}^{\left(\alpha \right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$
(3.20)

A recurrence formula for ${\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)$ is given by the next theorem.

Theorem 3.18 (Recurrence formula)

$\begin{array}{r}{\mathcal{F}}_{n+\nu }^{\left(\alpha +1\right)}\left(x;\lambda ;\mu ;\nu \right)\\ \phantom{\rule{1em}{0ex}}=\frac{{2}^{\mu }\left(n+1\right)\left(n+1\right)!}{\alpha \left(n+\nu \right)!}\left[\left(1-\frac{\alpha \nu }{n+1}\right){\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)+\left(\alpha -x\right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)\right].\end{array}$
(3.21)

Proof Setting

$g\left(t\right)={\left(\frac{\lambda {e}^{t}+1}{{2}^{\mu }{t}^{\nu }}\right)}^{\alpha }$

in (2.10), one can obtain

$\begin{array}{r}{\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)\\ \phantom{\rule{1em}{0ex}}=\left(x-\alpha +\frac{\alpha }{\lambda {e}^{t}+1}+\frac{\alpha \nu }{t}\right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)\\ \phantom{\rule{1em}{0ex}}=\left(x-\alpha \right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)+\frac{\alpha }{{t}^{\nu -1}}\cdot \frac{{t}^{\nu -1}}{\lambda {e}^{t}+1}{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)+\alpha \nu \cdot \frac{1}{t}{\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right).\end{array}$

By using Theorem 3.10 and (3.10), we have

$\begin{array}{rl}{\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)=& \left(x-\alpha \right){\mathcal{F}}_{n}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right)+\frac{\alpha \left(n+\nu \right)!}{{2}^{\mu }\left(n+1\right)\left(n+1\right)!}{\mathcal{F}}_{n+\nu }^{\left(\alpha +1\right)}\left(x;\lambda ;\mu ;\nu \right)\\ +\frac{\alpha \nu }{n+1}{\mathcal{F}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda ;\mu ;\nu \right).\end{array}$

After some calculations in the above equation, we get the desired result. □

Letting $\lambda ↦-\lambda$, taking $\mu =0$ and $\nu =1$ in (3.21) and noting relation (1.17), we deduce the following known result.

Corollary 3.19 (see, e.g., [, Remark 38])

${\mathcal{B}}_{n+1}^{\left(\alpha +1\right)}\left(x;\lambda \right)=\frac{1}{\alpha }\left[\left(\alpha -n-1\right){\mathcal{B}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda \right)+\left(n+1\right)\left(x-\alpha \right){\mathcal{B}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\right].$
(3.22)

Taking $\mu =1$ and $\nu =0$ in (3.21) and noting relation (1.18), we deduce the following known result.

Corollary 3.20 (see, e.g., [, Remark 39])

${\mathcal{E}}_{n}^{\left(\alpha +1\right)}\left(x;\lambda \right)=\frac{2{\left(n+1\right)}^{2}}{\alpha }\left[{\mathcal{E}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda \right)+\left(\alpha -x\right){\mathcal{E}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\right].$
(3.23)

Taking $\mu =\nu =1$ in (3.21) and noting relation (1.19), we deduce the following known result.

Corollary 3.21 (see, e.g., [, Remark 40])

${\mathcal{G}}_{n+1}^{\left(\alpha +1\right)}\left(x;\lambda \right)=\frac{2}{\alpha }\left[\left(n-\alpha +1\right){\mathcal{G}}_{n+1}^{\left(\alpha \right)}\left(x;\lambda \right)+\left(n+1\right)\left(\alpha -x\right){\mathcal{G}}_{n}^{\left(\alpha \right)}\left(x;\lambda \right)\right].$
(3.24)

Setting $\lambda =1$ in the above equation, we have the following.

Corollary 3.22 (see [, p.759, Theorem 4])

${G}_{n+1}^{\left(\alpha +1\right)}\left(x\right)=\frac{2}{\alpha }\left[\left(n-\alpha +1\right){G}_{n+1}^{\left(\alpha \right)}\left(x\right)+\left(n+1\right)\left(\alpha -x\right){G}_{n}^{\left(\alpha \right)}\left(x\right)\right].$
(3.25)

Taking $\lambda =-{\left(\frac{\beta }{a}\right)}^{b}$, $\mu =1-\kappa$, $\nu =\kappa$ in (3.21) and noting relation (1.21), thus we deduce the following result.

Corollary 3.23

$\begin{array}{rl}{\mathcal{Y}}_{n+\kappa ,\beta }^{\left(\alpha +1\right)}\left(x;\kappa ,a,b\right)=& \frac{{2}^{1-\kappa }\left(n+1\right)\left(n+1\right)!}{\alpha {a}^{b}\left(n+\kappa \right)!}\\ ×\left[\left(\frac{\alpha \kappa }{n+1}-1\right){\mathcal{Y}}_{n+1,\beta }^{\left(\alpha \right)}\left(x;k,a,b\right)+\left(x-\alpha \right){\mathcal{Y}}_{n,\beta }^{\left(\alpha \right)}\left(x;\kappa ,a,b\right)\right].\end{array}$
(3.26)

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## Acknowledgements

Dedicated to Professor Hari M. Srivastava on the occasion of his seventy-third birth anniversary. The authors sincerely thank the referees for their valuable suggestions and comments. The present investigation was supported by the National Natural Science Foundation of China under Grant 11226281, Fund of Science and Innovation of Yangzhou University, China under Grant 2012CXJ005, Natural Science Foundation Project of Chongqing, China under Grant CSTC2011JJA00024, Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625, Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07 and Program of Chongqing Innovation Team Project in University under Grant No. KJTD201308.

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Correspondence to Qiu-Ming Luo.

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The authors declare that they have no competing interests.

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All authors contributed equally in writing this paper, and read and approved the final manuscript.

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Lu, DQ., Xiang, CH. & Luo, QM. Some results for Apostol-type polynomials associated with umbral algebra. Adv Differ Equ 2013, 201 (2013). https://doi.org/10.1186/1687-1847-2013-201

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• DOI: https://doi.org/10.1186/1687-1847-2013-201

### Keywords

• generalized Apostol-type polynomials
• Sheffer sequences and Appell sequences
• umbral algebra
• Stirling numbers 