Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : q-Bernoulli numbers and q-Bernoulli polynomials revisited | Ryoo et al. Advances in Difference Equations 2011 2011 33 http content 2011 1 33 RESEARCH o Advances in Difference Equations a SpringerOpen Journal Open Access q-Bernoulli numbers and q-Bernoulli polynomials revisited Cheon Seoung Ryoo1 Taekyun Kim2 and Byungje Lee3 Correspondence ryoocs@ department of Mathematics Hannam University Daejeon 306791 Korea Full list of author information is available at the end of the article Abstract This paper performs a further investigation on the q-Bernoulli numbers and q-Bernoulli polynomials given by Acikgoz et al. Adv Differ Equ Article ID 951764 9 2010 some incorrect properties are revised. It is point out that the generating function for the q-Bernoulli numbers and polynomials is unreasonable. By using the theorem of Kim Kyushu J Math 48 73-86 1994 see Equation 9 some new generating functions for the q-Bernoulli numbers and polynomials are shown. Mathematics Subject Classification 2000 11B68 11S40 11S80 Keywords Bernoulli numbers and polynomials q-Bernoulli numbers and polynomials q-Bernoulli numbers and polynomials 1. Introduction As well-known definition the Bernoulli polynomials are given by ex eB x t Bn x ttn e 1 n n 0 see 1-4 with usual convention about replacing Bn x by Bn x . In the special case x 0 Bn 0 Bn are called the nth Bernoulli numbers. Let us assume that q e c with q 1 as an indeterminate. The q-number is defined by lxf 1 q 1 q see 1-6 . Note that limq 1 x q x. Since Carlitz brought out the concept of the q-extension of Bernoulli numbers and polynomials many mathematicians have studied q-Bernoulli numbers and q-Bernoulli polynomials see 1 7 5 6 8-12 . Recently Acikgoz Erdal and Araci have studied to a new approach to q-Bernoulli numbers and q-Bernoulli polynomials related to q-Bern-stein polynomials see 7 . But their generating function is unreasonable. The wrong properties are indicated by some counter-examples and they are corrected. It is point out that Acikgoz .
