A New Approach to q-Bernoulli Numbers and q-Bernoulli Polynomials Related to q-Bernstein Polynomials

Mehmet Açikgöz,Serkan Araci,Dilek Erdal

doi:10.1155/2010/951764

Abstract

We present a new generating function related to the -Bernoulli numbers and -Bernoulli polynomials. We give a new construction of these numbers and polynomials related to the second-kind Stirling numbers and -Bernstein polynomials. We also consider the generalized -Bernoulli polynomials attached to Dirichlet's character and have their generating function . We obtain distribution relations for the -Bernoulli polynomials and have some identities involving -Bernoulli numbers and polynomials related to the second kind Stirling numbers and -Bernstein polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of this generating function which interpolates the -Bernoulli polynomials at negative integers and is associated with -Bernstein polynomials.

Highlights

We present a new generating function related to the q-Bernoulli numbers and q-Bernoulli polynomials
We obtain distribution relations for the q-Bernoulli polynomials and have some identities involving q-Bernoulli numbers and polynomials related to the second kind Stirling numbers and q-Bernstein polynomials
We derive the q-extensions of zeta functions from the Mellin transformation of this generating function which interpolates the q-Bernoulli polynomials at negative integers and is associated with q-Bernstein polynomials

Summary

Introduction

We present a new generating function related to the q-Bernoulli numbers and q-Bernoulli polynomials. It is known that the Bernoulli polynomials are defined by et t − The recurrence formula for the classical Bernoulli numbers Bn is as follows, B0 1, B 1 n − Bn 0, if n > 0 The q-extension of the following recurrence formula for the Bernoulli numbers is

Results

Conclusion