-Genocchi Numbers and Polynomials Associated with -Genocchi-Type -Functions
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Review Article q-Genocchi Numbers and Polynomials Associated with q-Genocchi-Type l-Functions Yilmaz Simsek,1 Ismail Naci Cangul,2 Veli Kurt,1 and Daeyeoul Kim3 1
Department of Mathematics, Faculty of Arts and Science, University of Akdeniz, Antalya 07058, Turkey 2 Department of Mathematics, Faculty of Arts and Science, University of Uludag, Bursa 16059, Turkey 3 National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon 305-340, South Korea Correspondence should be addressed to Yilmaz Simsek, [email protected] Received 19 March 2007; Accepted 14 December 2007 Recommended by Rigoberto Medina The main purpose of this paper is to study generating functions of the q-Genocchi numbers and polynomials. We prove a new relation for the generalized q-Genocchi numbers, which is related to the q-Genocchi numbers and q-Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define q-Genocchi zeta and l-functions, which are interpolated q-Genocchi numbers and polynomials at negative integers. We also give some applications of the generalized q-Genocchi numbers. Copyright q 2008 Yilmaz Simsek 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 definitions and notations In 1, Jang et al. gave new formulae on Genocchi numbers. They defined poly-Genocchi numbers to give the relation between Genocchi numbers, Euler numbers, and poly-Genocchi numbers. In 2, Kim et al. constructed new generating functions of the q-analogue Eulerian numbers and q-analogue Genocchi numbers. They gave relations between Bernoulli numbers, Euler numbers, and Genocchi numbers. They also defined Genocchi zeta functions which interpolate these numbers at negative integers. Kim 3 gave new concept of the q-extension of Genocchi numbers and gave some relations between q-Genocchi polynomials and q-Euler numbers. In this paper, by using generating function of this numbers, we study q-Genocchi zeta and l-functions. In 4, Kim constructed q-Genocchi numbers and polynomials. By using these numbers and polynomials, he proved the q-analogue of alternating sums of powers of
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Advances in Difference Equations
consecutive integers due to Euler: k−1 j0
Gn,k,q − Gn,k,q k j : q2 −1j−1 jn−1 qk−jn1/2 1 qn
1.1
cf. 4, where if q ∈ C, |q| < 1, x x : q
1 − qx , 1−q
1 − q2j , j : q2 1 − q2
1.2
and the numbers Gn,k,q are called q-Genocchi numbers which are defined by 1 qt
∞
∞ tn qk−j j : q2 −1j−1 exp tj, q2 qk−j/2 Gn,k,q . n! j0 j0
1.3
Note that limq→1 x x, cf. 3, 5–9. The Euler numbers En are usually defined by means of the following generating function cf. 10–16: ∞ 2 tn , |t| < π. E n et 1 n0 n!
1.4
The Genocchi numbers Gn are usually defined by means of the following generating function cf. 12, 13: ∞
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