Some new formulas for the products of the Apostol type polynomials

Yuan He,Serkan Araci,HM Srivastava

doi:10.1186/s13662-016-1014-0

Abstract

In the year 2014, Kim et al. computed a kind of new sums of the products of an arbitrary number of the classical Bernoulli and Euler polynomials by using the Euler basis for the vector space of polynomials of bounded degree. Inspired by their work, in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler, and Genocchi polynomials by making use of the generating function methods and summation transform techniques. The results derived here are generalizations of the corresponding known formulas involving the classical Bernoulli, Euler, and Genocchi polynomials.

Highlights

The classical Bernoulli polynomials Bn(x), Euler polynomials En(x), and Genocchi polynomials Gn(x) are usually defined by the following generating functions: text ∞tn et – = Bn(x) n! |t| < π, ( . ) n= ext ∞tn et + = En(x) n!|t| < π, and text ∞tn et + = Gn(x) n! |t| < π
Motivated and inspired by the work of Kim et al [ ], in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler and Genocchi polynomials by making use of the generating function methods and summation transform techniques
Some known results for the classical Bernoulli, Euler, and Genocchi polynomials are shown to be derivable as special cases of our product formulas

Summary

Introduction

1 Introduction The classical Bernoulli polynomials Bn(x), Euler polynomials En(x), and Genocchi polynomials Gn(x) are usually defined by the following generating functions: text tn et – = Bn(x) n! Since the publication of the work by Luo and Srivastava [ – ], some interesting properties for the Apostol-Bernoulli, Euler and Genocchi polynomials have been well explored by many authors (see, for example, [ – ]).

Results

Conclusion