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Báo cáo hóa học: " Review Article On the Generalized q-Genocchi Numbers and Polynomials of Higher-Order"

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: Review Article On the Generalized q-Genocchi Numbers and Polynomials of Higher-Order | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 424809 8 pages doi 10.1155 2011 424809 Review Article On the Generalized q-Genocchi Numbers and Polynomials of Higher-Order C. S. Ryoo 1 T. Kim 2 J. Choi 2 and B. Lee3 1 Department of Mathematics Hannam University Daejeon 306-791 Republic of Korea 2 Division of General Education-Mathematics Kwangwoon University Seoul 139-701 Republic of Korea 3 Department of Wireless Communications Engineering Kwangwoon University Seoul 139-701 Republic of Korea Correspondence should be addressed to T. Kim tkkim@kw.ac.kr Received 10 August 2010 Accepted 18 February 2011 Academic Editor Roderick Melnik Copyright 2011 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. We first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to X. The purpose of this paper is to present a systemic study of some families of higher-order generalized q-Genocchi numbers and polynomials attached to X by using the generating function of those numbers and polynomials. 1. Introduction As a well known definition the Genocchi polynomials are defined by 2t ext eG x t ỳGn x in i n et . 1 e _ e AGnx . r n e n 0 n- 1.1 where we use the technical method s notation by replacing Gn x by Gn x symbolically see 1 2 . In the special case x 0 Gn Gn 0 are called the nth Genocchi numbers. From the definition of Genocchi numbers we note that G1 1 G3 G5 G7 0 and even coefficients are given by G2n 2 1 - 22n B2n 2nE2n-1 0 see 3 where Bn is a Bernoulli number and En x is an Euler polynomial. The first few Genocchi numbers for 2 4 6 . are -1 1 -3 17 -155 2073 . The first few prime Genocchi numbers are given by G6 -3 and G8 17. It is known that there are no other prime Genocchi numbers with n .