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 Relationships between the Analogs of Euler Numbers and Polynomials | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 86052 22 pages doi 2007 86052 Research Article Some Relationships between the Analogs of Euler Numbers and Polynomials C. S. Ryoo T. Kim and Lee-Chae Jang Received 5 June 2007 Revised 28 July 2007 Accepted 26 August 2007 Recommended by Narendra K. Govil We construct new twisted Euler polynomials and numbers. We also study the generating functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function twisted Hurwitz zeta function twisted Dirichlet -Euler numbers and twisted Euler polynomials at non-positive integers respectively. Furthermore we find distribution relations of generalized twisted Euler numbers and polynomials. By numerical experiments we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally we give a table for the solutions of the twisted q-Euler polynomials. Copyright 2007 C. S. Ryoo 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 and notations Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers Q denotes the field of rational numbers Qp denotes the field of p-adic rational numbers C denotes the complex numbers field and Cp denotes the completion of algebraic closure of Qp. Let Vp be the normalized exponential valuation of Cp with I p p p--Vp p p 1. When one talks of q-extension q is considered in many ways such as an indeterminate a complex number q e C or p-adic number q e Cp. If q e C one normally assumes that Iq I 1. If q e Cp we normally assume that Iq - 11p p-1 p-1 so that qx exp xlogq for x p 1. 1 - qx . x q x q 1-q cf. 1-18 . 1 - q 2 Journal of Inequalities and .
