Some new identities for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials

https://doi.org/10.1016/j.amc.2015.03.132Get rights and content

Abstract

In this paper, we present a further investigation for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. By making use of the generating function methods and summation transform techniques, we establish some new identities involving the products of the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. Many of the results presented here are the corresponding generalizations of some known formulas on the classical Bernoulli polynomials and the classical Genocchi polynomials.

Introduction

The classical Bernoulli polynomials Bn(x) and the classical Genocchi polynomials Gn(x) are usually defined by means of the following generating functions: textet1=n=0Bn(x)tnn!(|t|<2π)and 2textet+1=n=0Gn(x)tnn!(|t|<π).The rational numbers Bn and Gn given by Bn=Bn(0)andGn=Gn(0)are called the classical Bernoulli numbers and the classical Genocchi numbers, respectively. These polynomials and numbers play important roles in different areas of mathematics such as number theory, combinatorics, special functions and mathematical analysis. Numerous interesting properties for them can be found in many books and papers (see, for example, [2], [5], [6], [9], [12], [22], [24], [25], [42], [45], [46]). It is worth noticing that the classical Genocchi polynomials can be expressed in terms of the classical Bernoulli polynomials as follows: Gn(x)=2Bn(x)2n+1Bn(x2)(n0).

Some widely-investigated analogues of the classical Bernoulli polynomials Bn(x) and the classical Genocchi polynomials Gn(x) are the Apostol–Bernoulli polynomials Bn(x;λ) and Apostol–Genocchi polynomials Gn(x;λ), which are usually defined by means of the following generating functions (see, e.g., [30], [32], [33]): textλet1=n=0Bn(x;λ)tnn!(|t|<2πwhenλ=1;|t|<|logλ|whenλ1)and 2textλet+1=n=0Gn(x;λ)tnn!(|t|<πwhenλ=1;|t|<|log(λ)|whenλ1).In particular, Bn(λ) and Gn(λ) given by Bn(λ)=Bn(0;λ)andGn(λ)=Gn(0;λ)are called the Apostol--Bernoulli numbers and the Apostol–Genocchi numbers, respectively. Obviously, Bn(x;λ) and Gn(x;λ) reduce, respectively, to Bn(x) and Gn(x) when λ = 1. It is worth mentioning that the Apostol–Bernoulli polynomials were firstly introduced by Apostol [4] (see also Srivastava [46] for a systematic further study) in order to evaluate the value of the Hurwitz–Lerch zeta function. Since the publication of the works by Luo and Srivastava [32], [33], some interesting properties for the Apostol–Bernoulli and Apostol–Genocchi polynomials have been well explored by many authors. For example, Boyadzhiev [8] established some relationships between the Apostol–Bernoulli polynomials, the classical Eulerian polynomials and the derivative polynomials for the cotangent functions. Luo [29], [31] derived some multiplication formulas for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. Further, Luo [28], [30] gave the Fourier expansions for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials by applying the Lipschitz summation formula and obtained some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz (or generalized) zeta function. Tremblay et al. [47] investigated a new class of generalized Apostol–Bernoulli polynomials and obtained a generalization of the Srivastava–Pintér addition theorem (see [46]). Choi et al. [11] gave an explicit representation of the generalized Bernoulli polynomials in terms of a generalization for the Hurwitz–Lerch zeta function. Garg et al. [16] investigated some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch zeta functions (see also Lin et al. [26] for more general relationships). Kim and Hu [23] obtained the sums of products of any number of the Apostol–Bernoulli numbers which provide a generalization of the famous Euler’s formula for the classical Bernoulli numbers (see also He and Araci [20] for a further exploration for the sums of products of any number of the Apostol–Bernoulli polynomials following the work of Dilcher [15] as well as Kim and Hu [23]). More recently, Luo and Srivastava [34] introduced and systematically studied a unification of various Apostol type polynomials. See Lu and Srivastava [27] for some symmetric identities for these generalized Apostol type polynomials and Dere et al. [14], [38] for some unified presentations of generalized Apostol type polynomials (see also [40], [41], [43], [44] for several further developments on the subject).

Based upon p-adic arguments, Miki [36] discovered the following curious identity involving both an ordinary convolution and a binomial convolution of the classical Bernoulli numbers: k=2n2BkBnkk(nk)k=2n2(nk)BkBnkk(nk)=2HnBnn(n4),where Hn denotes the nth Harmonic numbers given by Hn=1+12++1n.Subsequently, Matiyasevich [35] obtained a companion identity to (1.6) with the aid of the computer software system Mathematica as follows: (n+2)k=2n2BkBnk2k=2n2(n+2k)BkBnk=n(n+1)Bn(n4).With the help of the Stirling numbers S(n, k) of the second kind, Gessel [17] extended Miki’s identity (1.7) to the classical Bernoulli polynomials. Gessel [17] also showed that, for every positive integer n≧2, k=1n1Bk(x)Bnk(x)k(nk)2nk=0n1(nk)Bk(x)BnknkBn1(x)=2Hn1Bn(x)n(n4),which was rederived by several authors using different methods (see, e.g., [7], [13], [21], [39]). More recently, by applying short and intelligible ideas, Agoh [3] rederived Gessel’s identity (1.8) and extended Matiyasevich’s identity (1.7) to hold true for the classical Bernoulli polynomials: k=1n1Bk(x)Bnk(x)2n+2k=0n1(n+2k)Bk(x)Bnk=n(n+1)6Bn1(x)+(n1)Bn(x)(n2).For some equivalent versions of (1.8), (see, e.g., [21], [39]). Further, Agoh [3] established four convolution identities involving the classical Bernoulli polynomials and the classical Genocchi polynomials similar to (1.8), (1.9). For example, Agoh [3] showed that, for every positive integer n≧2 , k=1n1Gk(x)Gnk(x)k(nk)+4nk=0n2(nk)Bk(x)Gnknk=0,and k=1n1Gk(x)Gnk(x)+4n+2k=0n2(n+2k)Bk(x)Gnk=0.

Motivated by the work of Agoh [3], in this paper we present a further investigation for the Apostol–Bernoulli polynomials Bn(x;λ) and the Apostol–Genocchi polynomials Gn(x;λ). We extend Agoh’s four convolution identities involving the classical Bernoulli polynomials Bn(x) and the classical Genocchi polynomials Gn(x) to hold true for the corresponding Apostol–Bernoulli polynomials Bn(x;λ) and the Apostol–Genocchi polynomials Gn(x;λ), respectively.

Our paper is organized as follows. In Section 2, we give some new formulas for the products of the Apostol–Bernoulli polynomials Bn(x;λ) and the Apostol–Genocchi polynomials Gn(x;λ). In Section 3, we extend Agoh’s four new convolution identities involving the classical Bernoulli polynomials Bn(x) and the classical Genocchi polynomials Gn(x) to hold true for the Apostol–Bernoulli polynomials Bn(x;λ) and the Apostol–Genocchi polynomials Bn(x;λ). Finally, our concluding remarks and observations are presented in the last section (Section 4).

Section snippets

Formulas involving the products of the Apostol type polynomials

In this section, we establish some new formulas for the products of the Apostol–Bernoulli polynomials Bn(x;λ) and the Apostol–Genocchi polynomials Gn(x;λ) by applying the generating function methods and summation transform techniques. For convenience, we denote by δ1, λ the Kronecker symbol given by δ1,λ={0(λ1)1(λ=1).We first state and prove the following result.

Theorem 1

Let m and n be positive integers. Then Gm(x;λ)Gn(y;μ)=2nk=0m(mk)[2k(xy)k1Gk(xy;λ)]Bm+nk(y;λμ)m+nk2mk=0n(nk)Gk(yx;μ)Bm+nk(x

Applications of Theorems 1 and 2

In this section, we make use of Theorems 1 and 2 to extend Agoh’s four convolution identities for the classical Bernoulli polynomials and the classical Genocchi polynomials to hold true for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials.

Theorem 3

Let n≧2 be a positive integer. Then k=1n1Gk(x;λ)Gnk(y;μ)k(nk)=4k=1n1(n1k)(xy)k1Bnk(y;λμ)nk2k=1n1(n1k)Gk(xy;λ)Bnk(y;λμ)+Gk(yx;μ)Bnk(x;λμ)k(nk)2δ1,λμ[1+(1)n]Gn(yx;1λ)n2.

Proof

Let m and n be positive integers. By

Concluding remarks and observations

We conclude this paper by remarking that the formula involving the products of the Apostol–Bernoulli polynomials can be expressed as follows (see, e.g., [18], [19], [48]): Bm(x;λ)Bn(y;μ)=nk=0m(mk)(k(xy)k1+Bk(xy;λ))Bm+nk(y;λμ)m+nk+mk=0n(nk)Bk(yx;μ)Bm+nk(x;λμ)m+nk+δ1,λμ(1)m+1·m!·n!·Bm+n(yx;1λ)(m+n)!(m,n1).Thus, by using the methods described in Section 3, one can easily find that, for every positive integer n ≧ 2, k=1n1Bk(x;λ)Bnk(y;μ)k(nk)=1nk=1n1(nk)(xy)k1Bnk(y;λμ)+1nk=1n1

Acknowledgments

This work is supported by the Foundation for Fostering Talents in Kunming University of Science and Technology (grant no. KKSY201307047) and the National Natural Science Foundation of the People’s Republic of China (grant no. 11326050).

References (48)

  • Q.-M. Luo et al.

    Some relationships between the Apostol–Bernoulli and Apostol–Euler polynomials

    Comput. Math. Appl.

    (2006)
  • Q.-M. Luo et al.

    Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind

    Appl. Math. Comput.

    (2011)
  • H. Miki

    A relation between Bernoulli numbers

    J. Number Theory

    (1978)
  • H. Ozden et al.

    Modification and unification of the Apostol-type numbers and polynomials and their applications

    Appl. Math. Comput.

    (2014)
  • H. Pan et al.

    New identities involving Bernoulli and Euler polynomials

    J. Combin. Theory Ser. A

    (2006)
  • H.M. Srivastava et al.

    Remarks on some relationships between the Bernoulli and Euler polynomials

    Appl. Math. Lett.

    (2004)
  • R. Tremblay et al.

    A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava–Pintér addition theorem

    Appl. Math. Lett.

    (2011)
  • M. Abramowitz et al.

    Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

    (1965)
  • T. Agoh

    Convolution identities for Bernoulli and Genocchi polynomials

    Electron. J. Combin.

    (2014)
  • T.M. Apostol

    On the Lerch zeta function

    Pacific J. Math.

    (1951)
  • I.V. Artamkin

    An elementary proof of the Miki–Zagier–Gessel identity

    Russ. Math. Surv.

    (2007)
  • K.N. Boyadzhiev

    Apostol–Bernoulli functions, derivative polynomials and Eulerian polynomials

    Adv. Appl. Discrete Math.

    (2008)
  • L. Carlitz

    Note on the integral of the product of several Bernoulli polynomials

    J. London Math. Soc.

    (1959)
  • J. Choi et al.

    A generalization of the Hurwitz–Lerch zeta function

    Integral Transforms Spec. Funct.

    (2008)
  • View full text