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New identities involving Bernoulli, Euler and Genocchi numbers

Su Hu1, Daeyeoul Kim2 and Min-Soo Kim3*

Author Affiliations

1 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea

2 National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon, 305-811, Republic of Korea

3 Division of Cultural Education, Kyungnam University, 7 (Woryeong-dong) kyungnamdaehak-ro, Masanhappo-gu, Changwon-si, Gyeongsangnam-do, 631-701, Republic of Korea

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Advances in Difference Equations 2013, 2013:74  doi:10.1186/1687-1847-2013-74

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/74


Received:26 October 2012
Accepted:5 March 2013
Published:26 March 2013

© 2013 Hu et al.; 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

Using p-adic integral, many new convolution identities involving Bernoulli, Euler and Genocchi numbers are given.

MSC: 11B68, 11S80.

Keywords:
Bernoulli numbers; Euler numbers; sums of products; fermionic p-adic integral; Volkenborn integral

1 Introduction

The study of the identities involving Bernoulli numbers and polynomials, Euler numbers and polynomials has a long history. More than 250 years ago, Euler discovered the following sums of products identity involving Bernoulli numbers:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M1">View MathML</a>

(1.1)

(see [1]).

This identity has been extended by many authors in different directions (see [2-21]). Recently, Kim and Hu [22] obtained an analogy of Euler’s sums of products identity for Apostol-Bernoulli numbers.

In 1978, Miki [23] obtained the following well-known identity, which is now known as Miki’s identity:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M2">View MathML</a>

for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M3">View MathML</a> , where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M4">View MathML</a>.

In 1997, Matiyasevich [24] found another identity of this type

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M5">View MathML</a>

for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M3">View MathML</a> .

In 2004, Dunne and Schubert [25] obtained the convolution identities for sums of products of Bernoulli numbers motivated by the role of these identities in quantum field theory and string theory. In 2006, Crabb [26] showed that Gessel’s generalization of Miki’s identity [27] is a direct consequence of a functional equation for the generating function. During the same year, Sun and Pan [28] established two general identities involving Bernoulli and Euler polynomials, which imply both Miki and Matiyasevich’s identities.

According to statement by Cohen in the first paragraph of [[29], Chapter 11], the p-adic functions with nice properties are powerful tools for studying many results of classical number theory in a straightforward manner, for instance, strengthening of almost all the arithmetic results on Bernoulli numbers.

For example, in Exercises 2 and 3 of [[29], Chapter 11], Cohen demonstrated a method to prove the following reciprocity formula for Bernoulli numbers using p-adic integral:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M7">View MathML</a>

(see Exercise 3(c) of [[29], Chapter 11]).

Recently, using p-adic integral, Kim et al.[8] proved several identities on Bernoulli and Euler numbers. More comprehensive coverage can be found in the monographs by Chio et al.[2], Kim et al.[4], Kim et al.[5], Kim et al.[6], Kim et al.[7], Kim and Kim [9], Kim et al.[13], Lee and Kim [14].

In this paper, following the methods of [8], we shall further provide many new convolution identities involving Bernoulli, Euler and Genocchi numbers.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M9">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M10">View MathML</a> be the nth Bernoulli, Euler and Genocchi polynomials, respectively. In what follows, we use <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M11">View MathML</a> to denote the special value of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M9">View MathML</a> at 0, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M13">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M14">View MathML</a> be the Kronecker symbol defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M15">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M16">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M17">View MathML</a>.

In fact, the following identities are shown in this paper:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M18">View MathML</a>

(1.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M19">View MathML</a>

(1.3)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M20">View MathML</a>

(1.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M21">View MathML</a>

(1.5)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M22">View MathML</a>

(1.6)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M23">View MathML</a>

(1.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M24">View MathML</a>

(1.8)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M25">View MathML</a>

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

This paper is organized as follows. In the next section, we recall the fundamental results between p-adic integral and Bernoulli and Euler numbers. Then using these results we prove new identities (identities (1.2), (1.3), (1.4), (1.5), (1.7), (1.8), (1.9)) involving Bernoulli and Euler numbers in Section 3, prove new identities (identities (1.10), (1.11), (1.12)) involving higher-order Bernoulli and Euler numbers and polynomials in Section 4, and also prove a new identity (identity (1.13)) involving both Bernoulli, Euler and Genocchi numbers in the final section.

2 p-adic analysis

In this section, we recall the fundamental results between p-adic integral and Bernoulli and Euler numbers, and we see that the properties of p-adic integrals may imply most well-known facts on Bernoulli numbers and polynomials, Euler numbers and polynomials.

We assume p is an odd prime number. The symbols <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M31">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M32">View MathML</a> denote the rings of p-adic integers, the field of p-adic numbers and the field of p-adic completion of the algebraic closure of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M31">View MathML</a>, respectively. ℕ denotes the set of natural numbers and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M34">View MathML</a> denotes <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M35">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M36">View MathML</a> be the space of uniformly (or strictly) differentiable function on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30">View MathML</a>. Then the Volkenborn integral of f is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M38">View MathML</a>

(2.1)

and this limit always exists when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M39">View MathML</a> (see [[30], p.264]). For such functions we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M40">View MathML</a>

(2.2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M41">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M42">View MathML</a>. By (2.2), we have the following Volkenborn p-adic integral representation of the generating function of Bernoulli polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M8">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M44">View MathML</a>

(2.3)

which is equivalent to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M45">View MathML</a>

(2.4)

(see [12]).

Setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46">View MathML</a> in (2.4), we obtain the Volkenborn p-adic integral representation of the nth Bernoulli numbers

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M47">View MathML</a>

(2.5)

(see [12,30]).

Thus from the binomial theorem and (2.4), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M48">View MathML</a>

(2.6)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M49">View MathML</a>

(2.7)

The recurrence formula of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M50">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M51">View MathML</a>

(2.8)

with the usual convention of denoting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M52">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M50">View MathML</a>.

By (2.6) and (2.8), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M54">View MathML</a>

(2.9)

The fermionic p-adic integral <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M57">View MathML</a>

(2.10)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M58">View MathML</a>. This integral was introduced by Kim [12] in order to derive useful formulas involving the Euler numbers and polynomials. It has also been defined independently by Shiratani and Yamamoto [31] in order to interpolate the Euler numbers p-adically. Osipov [32] gave a new proof of the existence of the Kubota-Leopoldt p-adic zeta function by using the integral representation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M59">View MathML</a>

(2.11)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M58">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M62">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M63">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M64">View MathML</a>. Note that when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M65">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M66">View MathML</a> is the fermionic p-adic integral <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30">View MathML</a>. Recently, the fermionic p-adic integral <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M30">View MathML</a> has been used by the third author to give a brief proof of Stein’s classical result on Euler numbers modulo power of two [10]; it has also been used by Maïga [33] to give some new identities and congruences concerning Euler numbers and polynomials.

From the definition of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M55">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M72">View MathML</a>

(2.12)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M42">View MathML</a> (see [[12], Lemma 1]). Thus from induction, we have the following fermionic p-adic integral equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M74">View MathML</a>

(2.13)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M76">View MathML</a>

(2.14)

(see [[12], Theorem 2]).

Using formula (2.12), we have the following fermionic p-adic integral representation of the generating function of Euler polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M9">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M78">View MathML</a>

(2.15)

namely

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M79">View MathML</a>

(2.16)

(see [[34], Proposition 2.1]).

Setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46">View MathML</a> in (2.16), we obtain the fermionic p-adic integral representation of the nth Euler numbers

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M81">View MathML</a>

(2.17)

(comparing with (2.5)).

Therefore, by the binomial theorem, (2.16), and (2.17), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M82">View MathML</a>

(2.18)

The recurrence formula of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M11">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M84">View MathML</a>

(2.19)

with the usual convention of denoting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M85">View MathML</a> by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M11">View MathML</a>.

By (2.18) and (2.19), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M87">View MathML</a>

(2.20)

The p-adic integral also implies the following reflection symmetric relations for the Bernoulli and the Euler polynomials (see formulas (2.12) and (2.13) in [34]):

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M88">View MathML</a>

(2.21)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M89">View MathML</a>

(2.22)

3 Identities involving Bernoulli and Euler numbers

In this section, we prove new identities involving Bernoulli and Euler numbers.

Lemma 3.1For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M90">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M91">View MathML</a>

Proof Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M92">View MathML</a> in (2.12), by (2.17), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M93">View MathML</a>

(3.1)

By (2.22), with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46">View MathML</a>, and (2.20), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M95">View MathML</a>

substituting it to (3.1), we have the desired result. □

The following well-known fact on Euler polynomials may also be established using p-adic integral. We refer to Corollary 1.1 in [35] for another proof.

Theorem 3.2<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M96">View MathML</a>for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M97">View MathML</a>.

Proof We prove our result following the method of the proof for Theorem 2.1 in [8].

By (2.17) and (2.18), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M98">View MathML</a>

(3.2)

On the other hand, using (2.18), (2.22) and Lemma 3.1, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M99">View MathML</a>

(3.3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M100">View MathML</a> is the Kronecker symbol.

Comparing (3.2) and (3.3), then using (2.18), with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M101">View MathML</a>, and (2.20), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M102">View MathML</a>

(3.4)

Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M103">View MathML</a> on the left-hand side of (3.4), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M104">View MathML</a>, hence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M105">View MathML</a>. Replacing n by 2n (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M97">View MathML</a>), the right-hand side becomes <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M96">View MathML</a>, which completes the proof. □

Theorem 3.3For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M109">View MathML</a>

Proof The proof goes the same way as in the above theorem.

By (2.5) and (2.6), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M110">View MathML</a>

(3.5)

On the other hand, using (2.6), (2.21) and Lemma 3.1, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M111">View MathML</a>

(3.6)

Comparing (3.5) and (3.6), then using (2.6), with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M101">View MathML</a>, and (2.9), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M113">View MathML</a>

Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M114">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M115">View MathML</a>) in this identity, we get our result. □

Theorem 3.4For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M117">View MathML</a>

(3.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M118">View MathML</a>

(3.8)

Taking <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M119">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M103">View MathML</a> in (3.7) and (3.8), respectively, we obtain the following corollary.

Corollary 3.5For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M122">View MathML</a>

(3.9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M123">View MathML</a>

(3.10)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M124">View MathML</a>

(3.11)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M125">View MathML</a>

(3.12)

Proof of Theorem 3.4 We prove our result following the method of the proof for Theorem 2.3 in [8].

Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M101">View MathML</a> in (2.6), (2.18), and using (2.9), (2.20), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M127">View MathML</a>

(3.13)

By the linear property of p-adic integral and (2.17), we have the following fermionic p-adic integral representation of the product of Bernoulli and Euler polynomials:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M128">View MathML</a>

(3.14)

Further, by (2.21), (2.22), Lemma 3.1 and (3.13), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M129">View MathML</a>

(3.15)

Comparing (3.14) and (3.15), then using (3.13), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M130">View MathML</a>

(3.16)

Replacing m by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M131">View MathML</a> and n by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M132">View MathML</a> in (3.16), respectively, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M133">View MathML</a>

(3.17)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M134">View MathML</a>

(3.18)

Finally, replacing n by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M135">View MathML</a> in (3.17) and m by 2m in (3.18), using Theorem 3.2, we get our result. □

Remark 3.6 Putting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M103">View MathML</a> in (3.17), we recover Theorem 3.3. Setting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M119">View MathML</a> in (3.18), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M138">View MathML</a>

This identity is trivial, because by Theorem 3.2, the <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M139">View MathML</a>th terms are identical to zero.

4 Identities involving higher-order Bernoulli and Euler numbers

In this section, we prove new identities involving higher-order Bernoulli and Euler numbers and polynomials.

The higher-order Bernoulli polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M140">View MathML</a> and the higher-order Euler polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M141">View MathML</a> are defined by the following generating functions:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M142">View MathML</a>

(4.1)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M143">View MathML</a>

(4.2)

respectively. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M145">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M146">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M147">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M148">View MathML</a> denote the higher-order Bernoulli and Euler numbers, respectively (see [[20], Section 1.6] and [[21], (16), (17)]). Moreover, from (4.1) and (4.2), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M149">View MathML</a>

(4.3)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M150">View MathML</a>

(4.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M151">View MathML</a>

(4.5)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M152">View MathML</a>

(4.6)

Lemma 4.1For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M90">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M155">View MathML</a>

Proof Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M156">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75">View MathML</a>) in (2.14), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M158">View MathML</a>.

By (2.13), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M159">View MathML</a>

(4.7)

By (2.20), (2.22) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46">View MathML</a>, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M161">View MathML</a>, and we obtain the assertion of the lemma. □

Theorem 4.2For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108">View MathML</a>, we have

Proof From (2.17), (4.3) and the linear property of p-adic integral, we obtain the following fermionic p-adic integral representation of the higher-order Bernoulli polynomials:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M165">View MathML</a>

(4.8)

On the other hand, by (4.4) and Lemma 4.1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M166">View MathML</a>

(4.9)

Clearly,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M167">View MathML</a>

(4.10)

Comparing (4.8) and (4.9), then using (4.10) and (4.3) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M168">View MathML</a>, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M169">View MathML</a>

(4.11)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M145">View MathML</a>, therefore, by (4.11), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M171">View MathML</a>

which is the required result. □

Theorem 4.3For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M108">View MathML</a>, we have

Proof This follows from the same process as in the proof of Theorem 4.2 by using (2.17), (4.5), (4.6) and Lemma 4.1. □

Theorem 4.4For<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M75">View MathML</a>, we have

Proof First we have the following fermionic p-adic integral representation of the product of higher-order Bernoulli and Euler polynomials:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M178">View MathML</a>

(4.12)

On the other hand, by (4.4), (4.6), Lemma 4.1, by (4.3) and (4.5) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M168">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M180">View MathML</a>

(4.13)

Comparing (4.12) and (4.13), and also noticing that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M145">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M183">View MathML</a>

which is the assertion of the theorem. □

5 Further remarks and observations

In this section, we show that the same methods as in Sections 3 and 4 can be used to obtain a new identity involving both Bernoulli, Euler and Genocchi numbers.

The higher-order Genocchi polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M184">View MathML</a> are defined by the generating function

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M185">View MathML</a>

(5.1)

The Genocchi polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M10">View MathML</a> are given by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M187">View MathML</a>. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46">View MathML</a>, we have the Genocchi numbers <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M189">View MathML</a>, i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M190">View MathML</a>. Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M46">View MathML</a>, we also have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M192">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M193">View MathML</a> denotes the higher-order Genocchi numbers.

The generating function of Genocchi polynomials is similar to those of Bernoulli and Euler polynomials, so it may be expected that the Genocchi numbers also satisfy similar identities as those established in Sections 3 and 4.

From the generating function (5.1), it is easy to deduce that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M194">View MathML</a>

(5.2)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M195">View MathML</a>

(5.3)

Therefore, by the fermionic p-adic integral representation of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M184">View MathML</a>, (5.2) and (5.3), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M197">View MathML</a>

(5.4)

For , by (5.4), we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M199">View MathML</a>. From (5.4), it is easily seen that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M200">View MathML</a>

(5.5)

Also, by the fermionic p-adic representations of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M140">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M184">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M203">View MathML</a>

(5.6)

which is equivalent to

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/74/mathml/M116">View MathML</a>.

Competing interests

The authors declare that they have no competing interest.

Authors’ contributions

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

Acknowledgements

This work was supported by the Kyungnam University Foundation Grant, 2013.

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