Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 956910, 8 pages doi:10.1155/2009/956910
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 General Education Department, Kookmin University, Seoul 136702, South Korea 3 Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139701, South Korea 2
Correspondence should be addressed to Kyung-Won Hwang,
[email protected] 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 padic q--series which interpolate q-Euler numbers at negative integers. Finally, we give Kummer Congruence for the q-extension of ordinary Euler numbers. Copyright q 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−vp p 1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or p-adic numbers q ∈ Cp . If q ∈ C, one normally assumes |q| < 1. If q ∈ Cp , one normally assumes |1 − q|p < 1. In this paper, we use the notations of q-number as follows see 1–37: 1 − qx , xq 1−q
x−q
x 1 − −q . 1q
1.1
The ordinary Euler numbers are defined as see 1–37 ∞ k0
Ek
tk 2 t , k! e 1
|t| < π,
1.2
2
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where 2/et 1 is written as eEt when Ek is replaced by Ek . From the definition of Euler number, we can derive E 1n En 0,
E0 1,
if n > 0,
1.3
with the usual convention of replacing Ei by Ei . Remark 1.1. The second kind Euler numbers are also defined as follows see 25: sech t
∞ k 2 2et ∗t E k et e−t e2t 1 k0 k!
|t|
0,
1.7
where Eqk must be replaced by Ek,q , symbolically. The q-Euler polynomials Ek,q x are given by qx Eq xq k , that is,
Ek,q x q Eq xq x
k
k k Ei,q qix xk−i q . i i0
1.8
Let d be a fixedodd positive integer. Then we have see 7 2q 2qd
dnq
d−1 a0
qa −1a En,q
x a d
En,q x,
for n ∈ Z .
1.9
We use 1.9 to get bounded p-adic q-Euler measures and finally take the Mellin transform to define p-adic q--series which interpolate q-Euler numbers at negative integers.
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2. p-adic q-Euler Measures Let d be a fixed odd positive integer, and let p be a fixed odd prime number. Define
Z X Xd lim , X1 Zp , ←− dpN Z N X∗ a dpZp ,
2.1
0
