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 Note on the q-Euler Measures | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 956910 8 pages doi 2009 956910 Research Article A Note on the q-Euler Measures Taekyun Kim 1 Kyung-Won Hwang 2 and Byungje Lee3 1 Division of General Education-Mathematics Kwangwoon University Seoul 139701 South Korea 2 General Education Department Kookmin University Seoul 136702 South Korea 3 Department of Wireless Communications Engineering Kwangwoon University Seoul 139701 South Korea Correspondence should be addressed to Kyung-Won Hwang khwang7@ Received 6 March 2009 Accepted 20 May 2009 Recommended by Patricia J. Y. Wong Properties of q-extensions of Euler numbers and polynomials which generalize those satisfied by Ek and Ek x are used to construct q-extensions of p-adic Euler measures and define p-adic q- -series which interpolate q-Euler numbers at negative integers. Finally we give Kummer Congruence for the q-extension of ordinary Euler numbers. Copyright 2009 Taekyun 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. 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 integers the field of p-adic rational numbers the complex number field and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with p p p V p 1 p. When one talks of q-extension q is variously considered as an indeterminate a complex number q e C or p-adic numbers q Cp. If q e C one normally assumes q 1. If q e Cp one normally assumes 1 - q p 1. In this paper we use the notations of q-number as follows see 1-37 x q 1 - qx ĩ-q x -q 1 - - x 1 q The ordinary Euler numbers are defined as see 1-37 _ tk 2 ZEkk. - ẽ ĩ 2 Advances in Difference Equations where 2 el 1 is written as eEt when Ek .
