Symmetric identities for Carlitz’s q-Bernoulli numbers and polynomials

Yuan He

doi:10.1186/1687-1847-2013-246

Abstract

In this paper, a further investigation for the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials is performed, and several symmetric identities for these numbers and polynomials are established by applying elementary methods and techniques. It turns out that various known results are derived as special cases.MSC:11B68, 05A19.

Highlights

The classical Bernoulli numbers Bn and Bernoulli polynomials Bn(x) are usually defined by the following generating functions t∞ tn et – = Bn n! |t| < π ( . ) n= and text ∞tn et – = Bn(x) n! |t| < π .Clearly, Bn = Bn( )
Throughout this paper, it is supposed that q ∈ C with |q| < and C being a complex number field
We recall the q-Bernoulli numbers βn = βn(q) and q-Bernoulli polynomials βn(x, q), which were introduced by Carlitz [ ], as follows

Summary

Introduction

Introduction The classical Bernoulli numbers Bn and Bernoulli polynomials Bn(x) are usually defined by the following generating functions t We will be concerned with the Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials. We recall the q-Bernoulli numbers βn = βn(q) and q-Bernoulli polynomials βn(x, q), which were introduced by Carlitz [ ], as follows The Carlitz’s q-Bernoulli numbers and q-Bernoulli polynomials can be defined by the following generating functions (see [ , ])

Objectives

Results

Conclusion