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 Some Identities on the Generalized q-Bernoulli Numbers and Polynomials Associated with q-Volkenborn Integrals | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 575240 17 pages doi 2010 575240 Research Article Some Identities on the Generalized q-Bernoulli Numbers and Polynomials Associated with q-Volkenborn Integrals T. Kim 1 J. Choi 1 B. Lee 2 and C. S. Ryoo3 1 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 2 Department of Wireless Communications Engineering Kwangwoon University Seoul 139-701 Republic of Korea 3 Department of Mathematics Hannam University Daejeon 306-791 Republic of Korea Correspondence should be addressed to T. Kim tkkim@ Received 23 August 2010 Accepted 30 September 2010 Academic Editor Alberto Cabada Copyright 2010 T. Kim 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 give some interesting equation of p-adic q-integrals on Zp. From those p-adic q-integrals we present a systemic study of some families of extended Carlitz type q-Bernoulli numbers and polynomials in p-adic number field. 1. Introduction Let p be a fixed prime number. Throughout this paper Zp Qp C and Cp will respectively denote the ring of p-adic rational integer the field of p-adic rational numbers the complex number field and the completion of algebraic closure of Qp. Let N be the set of natural numbers and Z 0 u N. Let vp be the normalized exponential valuation of Cp with p p p v- pi p . When one talks of q-extension q is considered as an indeterminate a complex number q e C or p-adic number q e Cp. If q e C we normally assume that q 1 and if q e Cp we normally assume that 1 - q p 1. We use the notation x q 1 - qx 1 - q. The q-factorial is defined as n q n q n - 1 q . 2 q 1 q 2 Journal of Inequalities and Applications and the Gaussian q-binomial coefficient is defined by n . n q Wq n - k q k q ln q n
