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

Open Access Research

Partial Hecke-type operators and their applications

Aykut Ahmet Aygunes and Yilmaz Simsek*

Author affiliations

Department of Mathematics, Faculty of Science, Akdeniz University, Campus, Antalya, 07058, Turkey

For all author emails, please log on.

Citation and License

Boundary Value Problems 2013, 2013:46  doi:10.1186/1687-2770-2013-46


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/46


Received:14 November 2012
Accepted:22 January 2013
Published:6 March 2013

© 2013 Aygunes and Simsek; 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

The aim of this paper is to give not only the matrix representation of partial Hecke-type operators by means of Bernoulli polynomials and Euler polynomials, but also functional equations and differential equations related to partial Hecke-type operators and special polynomials. By using these functional equations and differential equations, we derive some identities associated with special polynomials and partial Hecke-type operators. Moreover, we find several useful identities and relations using the partial Hecke operators.

MSC: 05B20, 11B68, 11F25.

Keywords:
Bernoulli polynomial; Euler polynomial; partial Hecke-type operator; total Hecke-type operator; Hurwitz zeta function; partial zeta function

1 Introduction

Recently, there have been many applications of Bernoulli polynomials and Euler polynomials in differential equations, in analytic number theory and in engineering. High-order linear differential-difference equations have also been solved in terms of Bernoulli polynomials. These polynomials are also related to several linear operators. In this paper, we investigate and derive several new identities related to the Hecke-type operators and generating functions for special polynomials.

Recently, many authors introduced and investigated the following generating functions which give us the Bernoulli polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M1">View MathML</a> and the Euler polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M2">View MathML</a>, respectively:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M3">View MathML</a>

(1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M4">View MathML</a>

(2)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M5">View MathML</a>, (1) and (2) are reduced to the generating functions for the Bernoulli numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M6">View MathML</a> and the Euler numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M7">View MathML</a>, respectively (cf.[1-4]), and see also the references cited in each of these earlier works.

The multiplication formulas for the Bernoulli and Euler polynomials are given as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M8">View MathML</a>

(3)

and for odd m,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M9">View MathML</a>

(4)

(cf.[5-8]), and see also the references cited in each of these earlier works.

The Bernoulli polynomials satisfy the following well-known identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M10">View MathML</a>

where m and n are positive integers (cf.[5,9,10]).

Bayad et al. [11] introduced and systematically studied the following family of partial Hecke-type operators on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M11">View MathML</a>.

Throughout this paper, we use the following notations: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M12">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M14">View MathML</a>.

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M16">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M17">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M18">View MathML</a>

Lemma 1.1 [[11], p.114, Lemma 1]

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M15">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M12">View MathML</a>, we have the following properties:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21">View MathML</a>preserves the degree in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M11">View MathML</a>.

(ii) By induction,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M23">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M24">View MathML</a>

(iii) For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M25">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M26">View MathML</a>be the canonical ℂ-basis of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M27">View MathML</a>

Then the matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M28">View MathML</a>corresponding to the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21">View MathML</a> (restricted to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M30">View MathML</a>) in the basis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M31">View MathML</a>is represented by:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M32">View MathML</a>

(5)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M33">View MathML</a>.

(iv) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M34">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M35">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M36">View MathML</a>

Consequently, for a given integern, there is only one monic polynomial<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M37">View MathML</a>with degreeninxsatisfying the functional equation (6).

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21">View MathML</a> satisfies the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M39">View MathML</a>

(6)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M12">View MathML</a>, from (6) and Lemma 1.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41">View MathML</a> is a monic polynomial (cf.[11]).

Remark 1.2 Equations (3) and (4) are closely related to the functional equation of (6). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M43">View MathML</a>, equation (6) is reduced to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M45">View MathML</a>, respectively. For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M15">View MathML</a>, we know that there is only one monic polynomial satisfying (6) by Lemma 1.1, and there already exist the functional equations as (3) and (4).

The total Hecke-type operators, associated with partial Hecke-type operators, are defined by Bayad et al. [[11], p.112, Eq. (1.6)] as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M47">View MathML</a>

Theorem 1.3[11]

Polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41">View MathML</a>are eigenfunctions for the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M49">View MathML</a>with eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M50">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M51">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M52">View MathML</a>is the Hurwitz zeta function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M53">View MathML</a>

2 Differential equations related to the partial Hecke-type operators and special polynomials

In this section, we derive some ordinary and partial differential equations not only for a generating function, but also for partial Hecke-type operators. We also give a functional equation for the generating function. We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M54">View MathML</a>

We now give an explicit formula of the generating function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M55">View MathML</a> as follows.

Theorem 2.1[11]

Generating functions for the polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41">View MathML</a>are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M57">View MathML</a>

The polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41">View MathML</a> are the so-called Bernoulli-Euler-type polynomials.

We derive the following partial differential equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M55">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M60">View MathML</a>

(7)

Theorem 2.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M61">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M62">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M63">View MathML</a>.

Proof By using (7), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M64">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M65">View MathML</a>

(8)

Therefore, by comparing the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M66">View MathML</a> on both sides of equation (8), we have the desired result.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42">View MathML</a>, we apply the same process. So, we omit it. □

We set the following differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M68">View MathML</a>

(9)

Theorem 2.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M69">View MathML</a>

Proof We make some arrangement (9) and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M70">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M71">View MathML</a>

From the above equation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M72">View MathML</a>

By comparing the coefficients of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M66">View MathML</a> on both sides of the above equation, we have the desired result. □

Remark 2.4 In Theorem 2.3, we obtain a convolution formula for the polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41">View MathML</a>. If we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M75">View MathML</a> into Theorem 2.3, then we get a convolution formula for the Eulerian-type numbers (cf.[10,13]).

Higher-order partial differential equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M76">View MathML</a> is given by the following theorem.

Theorem 2.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M64">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M61">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M79">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M80">View MathML</a>

Proof Taking vth derivative of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M76">View MathML</a>, with respect to x, we obtain the following higher-order partial differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M82">View MathML</a>

Using Theorem 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M83">View MathML</a>

Thus, we get the desired result. □

3 Matrix representations of partial Hecke-type operators

In this section, we give some numerical examples for the matrix representations of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M21">View MathML</a>. For the basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M85">View MathML</a>, our matrix representations contain Bernoulli polynomials and Euler polynomials for the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M87">View MathML</a>, respectively. Therefore, we need the following lemmas.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M89">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M90">View MathML</a>

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M88">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M89">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M93">View MathML</a>

Proofs of Lemma 3.1 and Lemma 3.2 have been given by many authors (among others) (cf.[2,4,8,10]).

In a special case, substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42">View MathML</a> into (iii) in Lemma 1.1 and using Lemma 3.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M95">View MathML</a>

According to the above equation, we are ready to give the main result of this section by the following theorem.

Theorem 3.3The matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M96">View MathML</a>corresponding to the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M86">View MathML</a> (restricted to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M30">View MathML</a>) in the basis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M99">View MathML</a>is represented by Bernoulli polynomials as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M100">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M43">View MathML</a> (iii) in Lemma 1.1 and using Lemma 3.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M102">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M103">View MathML</a>, then we obtain another main result by the following theorem.

Theorem 3.4Letabe an odd number. The matrix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M104">View MathML</a>corresponding to the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M87">View MathML</a> (restricted to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M30">View MathML</a>) in the basis<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M31">View MathML</a>is represented by Euler polynomials as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M108">View MathML</a>

4 Some applications of total Hecke-type operators

In this section, we give some applications related to eigenvalues for the total Hecke-type operators of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M110">View MathML</a>. We derive many new identities which are related not only to the total Hecke-type operators, but also to the Riemann zeta function, the Hurwitz zeta function, Bernoulli and Euler numbers, Euler identities and the convolution of Bernoulli and Euler numbers and polynomials.

Throughout this section, we use the following notation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M111">View MathML</a>

The partial zeta function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M112">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M113">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M115">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M116">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M117">View MathML</a>) (cf.[4,8,10,12,14]).

Theorem 4.1The polynomials<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M41">View MathML</a>are eigenfunctions for the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M49">View MathML</a>with eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M120">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M121">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M112">View MathML</a>is a partial zeta function.

Proof

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M123">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M124">View MathML</a>

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M127">View MathML</a> into the above equation, after using Theorem 1.3, we arrive at the desired result. □

Theorem 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M128">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M130">View MathML</a>

(10)

Proof Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M43">View MathML</a> in Theorem 1.3 and using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M132">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M133">View MathML</a>

(11)

We recall from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M134">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M135">View MathML</a> that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M136">View MathML</a>

(12)

(cf. [[10], p.96]). Combining (11) and (12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M137">View MathML</a>

(13)

If we replace n by 2n in the above equation, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M138">View MathML</a>

(14)

From the work of Srivastava and Choi [[4], p.98], we recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M139">View MathML</a>

(15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M128">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M142">View MathML</a>

(16)

By substituting (15) and (16) into (14), after some elementary calculations, we arrive at the desired result. □

Theorem 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M144">View MathML</a>

(17)

Proof Combining (14) and (16), we easily complete the proof of the theorem, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M145">View MathML</a>

 □

By using (10) and (17), we obtain a convolution formula (Euler identity) for Bernoulli numbers.

Theorem 4.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M147">View MathML</a>

(18)

Proof Since the left-hand sides of (10) and (17) are equal, the right-hand sides of (10) and (17) must be equal. Thus, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M148">View MathML</a>

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

Observe that the proof of (18) is also given in [4].

Theorem 4.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M150">View MathML</a>

Proof For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M152">View MathML</a>

(19)

(cf. [[4], p.131]). By using (14) and (19), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M153">View MathML</a>

Thus, the proof is completed. □

Theorem 4.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M155">View MathML</a>

(20)

Proof Consider that n is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M156">View MathML</a> in (13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M157">View MathML</a>

(21)

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>, one can easily get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M159">View MathML</a>

(22)

(cf. [[4], p.99, Eq. (21)]). Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M160">View MathML</a>

Thus, the proof is completed. □

Theorem 4.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M162">View MathML</a>

(23)

Proof Note that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M164">View MathML</a>

(24)

(cf. [[4], p.99, Eq. (22)]). By using (21) and (24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M165">View MathML</a>

Thus, the proof is completed. □

Theorem 4.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M167">View MathML</a>

Proof Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M42">View MathML</a> into Theorem 1.3 and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M170">View MathML</a>

(25)

If n is replaced by 2n in the above equation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M171">View MathML</a>

(26)

By using (16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M172">View MathML</a>

(27)

Thus, the proof is completed. □

Theorem 4.9Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M128">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M129">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M175">View MathML</a>

Proof By using (26), (15) and (16), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M176">View MathML</a>

Thus, the proof is completed. □

Theorem 4.10Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M178">View MathML</a>

Proof By using (26) and (19), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M179">View MathML</a>

Thus, the proof is completed. □

Theorem 4.11Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M181">View MathML</a>

(28)

Proof By replacing n by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M156">View MathML</a> in (25), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M183">View MathML</a>

(29)

By substituting (29) into (22), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M184">View MathML</a>

Thus, the proof is completed. □

Theorem 4.12Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M143">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M186">View MathML</a>

(30)

Proof By using (29) and (24), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M187">View MathML</a>

Thus, the proof is completed. □

By comparing (20) and (23) or (28) and (30), we arrive at the following result.

Corollary 4.13

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M188">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

Dedicated to Professor Hari M Srivastava.

All authors are partially supported by Research Project Offices Akdeniz Universities. We would like to thank the referees for their valuable comments.

References

  1. Ozden, H, Simsek, Y, Srivastava, HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl.. 60, 2779–2787 (2010). Publisher Full Text OpenURL

  2. Kim, T, Rim, S-H, Simsek, Y, Kim, D: On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions. J. Korean Math. Soc.. 45, 435–453 (2008). Publisher Full Text OpenURL

  3. Simsek, Y: Twisted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M189">View MathML</a>-Bernoulli numbers and polynomials related to twisted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/46/mathml/M190">View MathML</a>-zeta function and L-function. J. Math. Anal. Appl.. 324, 790–804 (2006). PubMed Abstract | Publisher Full Text OpenURL

  4. Srivastava, HM, Choi, J: Series Associated with the Zeta and Related Functions, Kluwer Academic, Dordrecht (2001)

  5. Carlitz, L: A note on the multiplication formulas for the Bernoulli and Euler polynomials. Proc. Am. Math. Soc.. 4, 184–188 (1953). Publisher Full Text OpenURL

  6. Luo, Q-M, Srivastava, HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl.. 308, 290–302 (2005). Publisher Full Text OpenURL

  7. Raabe, JL: Zurückführung einiger summen and bestimmten integrale auf die Jacob Bernoullische function. J. Reine Angew. Math.. 42, 348–376 (1851)

  8. Srivastava, HM, Kim, T, Simsek, Y: q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys.. 12, 241–268 (2005)

  9. Lehmer, DH: A new approach to Bernoulli polynomials. Am. Math. Mon.. 95, 905–911 (1998)

  10. Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdam (2012)

  11. Bayad, A, Aygunes, AA, Simsek, Y: Hecke operators and generalized Bernoulli-Euler polynomials. J. Algebra Number Theory, Adv. Appl.. 3, 111–122 (2010)

  12. Whittaker, ET, Watson, GN: A Course of Modern Analysis, Cambridge University Press, Cambridge (1962)

  13. Chu, W, Zhou, RR: Convolutions of Bernoulli and Euler polynomials. Sarajevo J. Math.. 6, 147–163 (2010)

  14. Washington, LC: Introduction to Cyclotomic Fields, Springer, Berlin (1982)