Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article A Note on Symmetric Properties of the Twisted q-Bernoulli Polynomials and the Twisted Generalized q-Bernoulli Polynomials | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 801580 13 pages doi 2010 801580 Research Article A Note on Symmetric Properties of the Twisted q-Bernoulli Polynomials and the Twisted Generalized q-Bernoulli Polynomials . Jang 1 H. Yi 2 K. Shivashankara 3 T. Kim 4 Y. H. Kim 4 and B. Lee5 1 Department of Mathematics and Computer Science KonKuk University Chungju 138-70 Republic of Korea 2 Department of Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 3 Department of Mathematics Yuvaraja s College University of Mysore Mysore 570 005 India 4 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 5 Department of Wireless Communications Engineering Kwangwoon University Seoul 139-701 Republic of Korea Correspondence should be addressed to H. Yi hsyi@ Received 11 September 2009 Revised 14 April 2010 Accepted 31 May 2010 Academic Editor Abdelkader Boucherif Copyright 2010 . Jang 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 define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomials attached to X of higher order and investigate some symmetric properties of them. Furthermore using these symmetric properties of them we can obtain some relationships between twisted q-Bernoulli numbers and polynomials and between twisted generalized q-Bernoulli numbers and polynomials. 1. Introduction Let p be a fixed prime number. Throughout this paper Zp Qp and Cp will respectively denote the ring of p-adic rational integers the field of p-adic rational numbers and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with p p p Vp p p-1. When one talks of q-extension q is variously considered as an indeterminate a complex .
