Some new identities for the Apostol–Bernoulli polynomials and the Apostol–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: and The rational numbers Bn and Gn given by 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:
Some widely-investigated analogues of the classical Bernoulli polynomials Bn(x) and the classical Genocchi polynomials Gn(x) are the Apostol–Bernoulli polynomials and Apostol–Genocchi polynomials which are usually defined by means of the following generating functions (see, e.g., [30], [32], [33]): and In particular, and given by are called the Apostol--Bernoulli numbers and the Apostol–Genocchi numbers, respectively. Obviously, and 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: where Hn denotes the nth Harmonic numbers given by Subsequently, Matiyasevich [35] obtained a companion identity to (1.6) with the aid of the computer software system Mathematica as follows: 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, 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: 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 , and
Motivated by the work of Agoh [3], in this paper we present a further investigation for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials . 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 and the Apostol–Genocchi polynomials respectively.
Our paper is organized as follows. In Section 2, we give some new formulas for the products of the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials . 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 and the Apostol–Genocchi polynomials . 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 and the Apostol–Genocchi polynomials by applying the generating function methods and summation transform techniques. For convenience, we denote by δ1, λ the Kronecker symbol given by
We first state and prove the following result.
Theorem 1 Let m and n be positive integers. Then
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
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]): Thus, by using the methods described in Section 3, one can easily find that, for every positive integer n ≧ 2,
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).
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