Some results for Carlitz’s q-Bernoulli numbers and polynomials

Yuan He

doi:10.2298/aadm140626006h

Abstract

A further investigation for Carlitz?s q-Bernoulli numbers and polynomials is performed, and several new formulae for these numbers and polynomials are established by applying some summation transform techniques. Special cases as well as immediate consequences of the main results are also presented.

Highlights

The classical Bernoulli polynomials Bn(x) are usually defined by the following exponential generating function: (1.1) text et − 1 = ∞ Bn(x) tn n! n=0 (|t| < 2π).In particular, the rational numbers Bn = Bn(0) are called the classical Bernoulli numbers
Numerous interesting properties for them can be found in many books; see, for example, [9, 23, 30])
We consider Carlitz’s q-Bernoulli numbers βn(q) and q-Bernoulli polynomials βn(x, q), which are respectively given by means of

Summary

Introduction

The classical Bernoulli polynomials Bn(x) are usually defined by the following exponential generating function: (1.1) Carlitz’s q-Bernoulli numbers and polynomials can be defined by the following exponential generating functions (see, e.g., [24, 27]): (1.5) Observe that for non-negative integers n, k, r, n+k+r i Notice that for a complex number s and non-negative integers p, h (cf the identity of Wu described in [10, 14, 29]), (2.9)

Results

Conclusion