Some results for the q-Bernoulli, q-Euler numbers and polynomials

Daeyeoul Kim,Min-Soo Kim

doi:10.1186/1687-1847-2011-68

Abstract

The q-analogues of many well known formulas are derived by using several results of q-Bernoulli, q-Euler numbers and polynomials. The q-analogues of ζ-type functions are given by using generating functions of q-Bernoulli, q-Euler numbers and polynomials. Finally, their values at non-positive integers are also been computed.

Highlights

Carlitz [1,2] introduced q-analogues of the Bernoulli numbers and polynomials
By using generating functions of q-Bernoulli, q-Euler numbers, and polynomials, we present the q-analogues of ζ-type functions
2. q-Bernoulli, q-Euler numbers and polynomials related to the Bosonic and the Fermionic p-adic integral on Zp we provide some basic formulas for p-adic q-Bernoulli, p-adic q-Euler numbers and polynomials which will be used to prove the main results of this article

Summary

Introduction

Carlitz [1,2] introduced q-analogues of the Bernoulli numbers and polynomials. From that time on these and other related subjects have been studied by various authors (see, e.g., [3,4,5,6,7,8,9,10]). It is well known that the Bernoulli numbers can be expressed as follows pN −1 ak a=0 (1:3) We derive q-analogues of many well known formulas by using several results of q-Bernoulli, q-Euler numbers, and polynomials. By using generating functions of q-Bernoulli, q-Euler numbers, and polynomials, we present the q-analogues of ζ-type functions.

Results

Conclusion