Advances in Difference Equations
Volume 2010 (2010), Article ID 951764, 9 pages
Research Article

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

Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, 27310 Gaziantep, Turkey

Received 24 November 2010; Accepted 27 December 2010

Academic Editor: Claudio Cuevas

Copyright © 2010 Mehmet Açikgöz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a new generating function related to the 𝑞 -Bernoulli numbers and 𝑞 -Bernoulli polynomials. We give a new construction of these numbers and polynomials related to the second-kind Stirling numbers and 𝑞 -Bernstein polynomials. We also consider the generalized 𝑞 -Bernoulli polynomials attached to Dirichlet's character 𝜒 and have their generating function. 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. Finally, 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.