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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Some properties of the generalized Apostol-type polynomials

Da-Qian Lu1 and Qiu-Ming Luo2*

Author Affiliations

1 Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu, 225002, People’s Republic of China

2 Department of Mathematics, Chongqing Normal University, Chongqing Higher Education Mega Center, Huxi Campus, Chongqing, 401331, People’s Republic of China

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Boundary Value Problems 2013, 2013:64  doi:10.1186/1687-2770-2013-64

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/64


Received:10 December 2012
Accepted:19 March 2013
Published:28 March 2013

© 2013 Lu and Luo; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Keywords:
generalized Apostol type polynomials; recurrence relations; differential equations; connected problems; quasi-monomial

1 Introduction, definitions and motivation

The classical Bernoulli polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M1">View MathML</a>, the classical Euler polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M2">View MathML</a> and the classical Genocchi polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M3">View MathML</a>, together with their familiar generalizations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M6">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M9">View MathML</a>

(1.3)

So that, obviously, the classical Bernoulli polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M1">View MathML</a>, the classical Euler polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M2">View MathML</a> and the classical Genocchi polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M3">View MathML</a> are given, respectively, by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M13">View MathML</a>

(1.4)

For the classical Bernoulli numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M14">View MathML</a>, the classical Euler numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M15">View MathML</a> and the classical Genocchi numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M16">View MathML</a> of order n, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M17">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M18">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>) are defined by means of the following generating function:

(1.6)

with, of course,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M21">View MathML</a>

(1.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M22">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M23">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>) of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25">View MathML</a> are defined by means of the following generating function:

(1.8)

with, of course,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M27">View MathML</a>

(1.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M28">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M29">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>) of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M32">View MathML</a>

(1.10)

with, of course,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M33">View MathML</a>

(1.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M34">View MathML</a> denotes the so-called Apostol-Euler numbers of order α.

On the subject of the Genocchi polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M3">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M36">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>) of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M39">View MathML</a>

(1.12)

with, of course,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M40">View MathML</a>

(1.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M43">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M44">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M46">View MathML</a>) of order α are defined by means of the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M47">View MathML</a>

(1.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M48">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M52">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M54">View MathML</a>) is said to be a quasi-monomial [16], whenever two operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M56">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M59">View MathML</a>

(1.21)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M60">View MathML</a>

(1.22)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M61">View MathML</a>

(1.23)

is analytic function at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M62">View MathML</a>.

From [18], we know that the multiplicative and derivative operators of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M63">View MathML</a> are

(1.24)

(1.25)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M66">View MathML</a>

(1.26)

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

Lemma 1.6 ([18])

The Appell polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M63">View MathML</a>defined by (1.22) satisfy the differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M68">View MathML</a>

(1.27)

where the numerical coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M70">View MathML</a>are defined in (1.26), and are linked to the values<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M71">View MathML</a>by the following relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M72">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M73">View MathML</a> be the vector space of polynomials with coefficients in ℂ. A polynomial sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M74">View MathML</a> be a polynomial set. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M74">View MathML</a> is called a σ-Appell polynomial set of transfer power series A is generated by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M76">View MathML</a>

(1.28)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M77">View MathML</a> 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M79">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M80">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M81">View MathML</a>be twoσ-Appell polynomial sets of transfer power series, respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M82">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M83">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M84">View MathML</a>

(1.29)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M85">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M86">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M87">View MathML</a>

(2.1)

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

Theorem 2.1For any integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90">View MathML</a>, the following recurrence relation for the generalized Apostol type polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M44">View MathML</a>holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M92">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 2.1, and then multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96">View MathML</a> on both sides of the result, we have:

Corollary 2.2For any integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90">View MathML</a>, the following recurrence relation for the generalized Apostol-Bernoulli polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M100">View MathML</a>holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M101">View MathML</a>

(2.3)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103">View MathML</a> in Theorem 2.1, we have the following corollary.

Corollary 2.3For any integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90">View MathML</a>, the following recurrence relation for the generalized Apostol-Euler polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M29">View MathML</a>holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M108">View MathML</a>

(2.4)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 2.1, we have the following corollary.

Corollary 2.4For any integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90">View MathML</a>, the following recurrence relation for the generalized Apostol-Genocchi polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M36">View MathML</a>holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M115">View MathML</a>

(2.5)

From (1.14) and (1.22), we know that the generalized Appostol type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M86">View MathML</a> is Appell polynomials with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M117">View MathML</a>

(2.6)

From the Eq. (23) of [15], we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M118">View MathML</a>. So from (2.6) and (1.12), we can obtain that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M120">View MathML</a>

(2.7)

By using (1.24) and (1.26), we can obtain the multiplicative and derivative operators of the generalized Appostol type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M44">View MathML</a>

(2.8)

(2.9)

From (2.1), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M124">View MathML</a>

(2.10)

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

Theorem 2.5For any integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90">View MathML</a>, the following recurrence relation for the generalized Apostol type polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M128">View MathML</a>holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M129">View MathML</a>

(2.11)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> in Theorem 2.5, we have the following corollary.

Corollary 2.6For any integral<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M19">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M90">View MathML</a>, the following recurrence relation for the generalized Apostol-Euler polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M134">View MathML</a>holds true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M135">View MathML</a>

(2.12)

Furthermore, applying Lemma 1.7 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M136">View MathML</a>, we have the following theorem.

Theorem 2.7The generalized Apostol type polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M136">View MathML</a>satisfy the differential equation:

(2.13)

Specially, by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> in Theorem 2.7, then we have the following corollary.

Corollary 2.8The generalized Apostol-Euler polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M134">View MathML</a>satisfy the differential equation:

(2.14)

3 Connection problems

From (1.14) and (1.28), we know that the generalized Apostol type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M142">View MathML</a> are a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M143">View MathML</a>-Appell polynomial set, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M143">View MathML</a> denotes the derivative operator.

From Table 1 in [19], we know that the derivative operators of monomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M145">View MathML</a> and the Gould-Hopper polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M146">View MathML</a>[30] are all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M143">View MathML</a>. And their transfer power series <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M148">View MathML</a> are 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M149">View MathML</a>, respectively.

Applying Lemma 1.7 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151">View MathML</a>, we have the following theorem.

Theorem 3.1

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M152">View MathML</a>

(3.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M153">View MathML</a>is the so-called Apostol type numbers of orderαdefined by (1.15).

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.1, and then multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96">View MathML</a> on both sides of the result, we have the following corollary.

Corollary 3.2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M158">View MathML</a>

(3.2)

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

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103">View MathML</a> in Theorem 3.1, we have the following corollary.

Corollary 3.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M161">View MathML</a>

(3.3)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.1, we have the following corollary.

Corollary 3.4

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M164">View MathML</a>

(3.4)

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

Applying Lemma 1.7 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151">View MathML</a>, we have the following theorem.

Theorem 3.5

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M167">View MathML</a>

(3.5)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M153">View MathML</a>is the so-called Apostol type numbers of orderαdefined by (1.15).

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.5, and then multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96">View MathML</a> on both sides of the result, we have the following corollary.

Corollary 3.6

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M173">View MathML</a>

(3.6)

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

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103">View MathML</a> in Theorem 3.5, we have the following corollary.

Corollary 3.7

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M176">View MathML</a>

(3.7)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.5, we have the following corollary.

Corollary 3.8

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M179">View MathML</a>

(3.8)

Applying Lemma 1.7 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151">View MathML</a>, we have the following theorem.

Theorem 3.9

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M182">View MathML</a>

(3.9)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.9, and then multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96">View MathML</a> on both sides of the result, we have the following corollary.

Corollary 3.10

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M187">View MathML</a>

(3.10)

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

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M103">View MathML</a> in Theorem 3.9, we have the following corollary.

Corollary 3.11

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M190">View MathML</a>

(3.11)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.9, we have the following corollary.

Corollary 3.12

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M193">View MathML</a>

(3.12)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M194">View MathML</a>, applying Lemma 1.7 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M151">View MathML</a>, we have the following theorem.

Theorem 3.13If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M197">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M198">View MathML</a>

(3.13)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.13, and then multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96">View MathML</a> on both sides of the result, we have the following corollary.

Corollary 3.14

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M203">View MathML</a>

(3.14)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.13, we have the following corollary.

Corollary 3.15

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M206">View MathML</a>

(3.15)

which is just the case of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M207">View MathML</a>in (3.4).

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M209">View MathML</a>, applying Lemma 1.7 to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M211">View MathML</a>, we can obtain the following theorem.

Theorem 3.16If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M209">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M214">View MathML</a>

(3.16)

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M94">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.13, and then multiplying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M96">View MathML</a> on both sides of the result, we have the following corollary.

Corollary 3.17

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M219">View MathML</a>

(3.17)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M207">View MathML</a> in (3.17), it is just (3.15).

By setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M95">View MathML</a> in Theorem 3.16, we have the following corollary.

Corollary 3.18

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M223">View MathML</a>

(3.18)

which is equal to (3.8).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M209">View MathML</a> in Theorem 3.16, we have:

Corollary 3.19

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M225">View MathML</a>

(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) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M226">View MathML</a> are defined by the series [31]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M227">View MathML</a>

(4.1)

with the following generating function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M228">View MathML</a>

(4.2)

And the 2VHKdFP <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M226">View MathML</a> are also defined through the operational identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M230">View MathML</a>

(4.3)

Acting the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M231">View MathML</a> on (1.14), and by the identity [32]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M232">View MathML</a>

(4.4)

we define the Hermite-based generalized Apostol type polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M233">View MathML</a> by the generating function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M234">View MathML</a>

(4.5)

Clearly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M235">View MathML</a>

From the generating function (4.5), we easily obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M236">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M237">View MathML</a>

(4.7)

which can be combined to get the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M238">View MathML</a>

(4.8)

Acting with the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M239">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/64/mathml/M245">View MathML</a> are the Hermite-based generalized Apostol-Euler polynomials and the Hermite-based generalized Apostol-Genocchi polynomials respectively, defined by the following generating functions:

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The present investigation was supported, in part, by the National Natural Science Foundation of China under Grant 11226281, Fund of Science and Innovation of Yangzhou University, China under Grant 2012CXJ005, Research Project of Science and Technology of Chongqing Education Commission, China under Grant KJ120625 and Fund of Chongqing Normal University, China under Grant 10XLR017 and 2011XLZ07.

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