Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A New Approach to q-Bernoulli Numbers and q-Bernoulli Polynomials Related to q-Bernstein Polynomials | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 951764 9 pages doi 2010 951764 Research Article A New Approach to q-Bernoulli Numbers and q-Bernoulli Polynomials Related to q-Bernstein Polynomials Mehmet Acikgoz Dilek Erdal and Serkan Araci Department of Mathematics Faculty of Science and Arts University of Gaziantep 27310 Gaziantep Turkey Correspondence should be addressed to Mehmet Acikgoz acikgoz@ Received 24 November 2010 Accepted 27 December 2010 Academic Editor Claudio Cuevas Copyright 2010 Mehmet Acikgoz 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 X 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. 1. Introduction Definitions and Notations Let C be the complex number field. We assume that q e C with q 1 and that the q-number is defined by x q q - 1 q - 1 in this paper. Many mathematicians have studied q-Bernoulli q-Euler polynomials and related topics see 1-23 . It is known that the Bernoulli polynomials are defined by t tn __ext y R X et - 1e tJBn X n n 0 for t 2n and that Bn Bn 0 are .
