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: Some new identities on the twisted carlitz’s q-bernoulli numbers and q-bernstein polynomials | Jang et al. Journal of Inequalities and Applications 2011 2011 52 http content 2011 1 52 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Some new identities on the twisted carlitz s q-bernoulli numbers and q-bernstein polynomials Lee-Chae Jang 1 Taekyun Kim2 Young-Hee Kim2 and Byungje Lee3 Correspondence tkkim@ 2Division of General EducationMathematics Kwangwoon University Seoul 139-701 Republic of Korea Full list of author information is available at the end of the article Abstract In this paper we consider the twisted Carlitz s q-Bernoulli numbers using p-adic flintegral on Zp. From the construction of the twisted Carlitz s q-Bernoulli numbers we investigate some properties for the twisted Carlitz s q-Bernoulli numbers. Finally we give some relations between the twisted Carlitz s q-Bernoulli numbers and q-Bernstein polynomials. Keywords q-Bernoulli numbers p-adic q-integral twisted 1. Introduction and preliminaries Let p be a fixed prime number. Throughout this paper Zp Qp and Cp will denote the ring of Ji-adic integers the field of Ji-adic rational numbers and the completion of algebraic closure of Qp respectively. Let N be the set of natural numbers and let Z N u 0 . Let vp be the normalized exponential valuation of Cp with p p p-vp p p. In this paper we assume that q e Cp with 1 - q p 1. The q-number is defined by Mq 1 qx 1 - q . Note that limq 1 x q x. We say that f is a uniformly differentiable function at a point a e Zp and denote this property by f e UD Zp if the difference quotient Ff x y f Wf has a limit f a as x y a a . Forf e UD Zp the Ji-adic q-integral on Zp which is called the q-Volkenborn integral is defined by Kim as follows hf f x dpq x lim f x qx see 1 - 1 N- pN q x o Zp In 2 Carlitz defined q-Bernoulli numbers which are called the Carlitz s q-Bernoulli numbers by Ao q 1 and qq 1 n - a q 11 if n 1 2 with the usual convention about replacing b by b q. In 2 3
