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Báo cáo hóa học: " Research Article A Note on the q-Genocchi Numbers and Polynomials"

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-Genocchi Numbers and Polynomials | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 71452 8 pages doi 10.1155 2007 71452 Research Article A Note on the g-Genocchi Numbers and Polynomials Taekyun Kim Received 15 March 2007 Revised 7 May 2007 Accepted 24 May 2007 Recommended by Paolo Emilio Ricci We discuss new concept of the g-extension of Genocchi numbers and give some relations between g-Genocchi polynomials and g-Euler numbers. Copyright 2007 Taekyun Kim. 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 The Genocchi numbers Gn n 0 1 2 . which can be defined by the generating function -2 s Gn t n 1.1 e 1 n0 n have numerous important applications in number theory combinatorics and numerical analysis among other areas 1-13 . It is easy to find the values G1 1 G3 G5 G7 0 and even coefficients are given by G2m 2 1 - 22n B2n 2nE2n-1 0 where Bn is a Bernoulli number and En x is an Euler polynomial. The first few Genocchi numbers for n 2 4 . are -1 -3 17 -155 2073 . The Euler polynomials are well known as 2 .A tn . -2-ext Y En x L- see 1 3 7-9 . 1.2 e 1 n0 n By 1.1 and 1.2 we easily see that E x - Í ữnrx-k where fi n n - 1 -- - k 1 cf. 4-6 . k 0 1.3 2 Journal of Inequalities and Applications For m n 1 and m odd we have nm - n Gm 1 m nkGkZm-k n - 1 1.4 k 1 K where Zm n 1m - 2m 3m - -1 n-1nm see 3 13 . From 1.15 we derive 2t z G 1 n Gn 1.5 n 0 n where we use the technique method notation by replacing Gm by Gm m 0 symbolically. By comparing the coefficients on both sides in 1.5 we see that G0 0 G 1 n Gn - if n 1 if n 1. 1.6 2 0 Let p be a fixed odd prime and let Cp denote the p-adic completion of the algebraic closure of Qp p-adic number field . For d is a fixed positive integer with p d 1 let X Xd lim z . N dpN Z X1 Zp X u a dpZp 1.7 0 a dp a p 1 a dpNZp x e X I x a mod d pN where a e Z .