Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2008, Article ID 304539, 13 pages doi:10.1155/2008/304539 Research Article Multivariate p-adic Fermionic -Integral on Z p and Related Multiple Zeta-Type Functions Min-Soo Kim, 1 Taekyun Kim, 2 and Jin-Woo Son 1 1 Department of Mathematics, Kyungnam University, Masan 631-701, South Korea 2 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea Correspondence should be addressed to Min-Soo Kim, mskim@kyungnam.ac.kr Received 14 April 2008; Accepted 27 May 2008 Recommended by Ferhan Atici In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz s type - Bernoulli polynomials and obtained the distribution relation for them. They also raised the following problem: are there analytic multiple twisted Carlitz s type -zeta functions which interpolate multiple twisted Carlitz s type -Euler Bernoulli polynomials? The aim of this paper is to give a partial answer to this problem. Furthermore we derive some interesting identities related to twisted - extension of Euler polynomials and multiple twisted Carlitz s type -Euler polynomials. Copyright 2008 Min-Soo 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, definitions, and notations Let p be an odd prime. Z p, Q p, and C p will always denote, respectively, the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of Q p. Let v p : C p Q { }Q is the field of rational numbers denote the p-adic valuation of C p normalized so that v p p 1. The absolute value on C p will be denoted as p, and x p p v px for x C p. We let Z p {x Z p 1/x Z p }.Ap-adic integer in Z p is sometimes called a p-adic unit. For each integer N 0, C p N will denote the multiplicative group of the primitive p N th roots of unity in C p C p \{0}. Set T p { ω C p ω pn 1 for some N 0 } N 0C p N. 1.1 The dual of Z p, in the sense of p-adic Pontrjagin duality, is T p C p, the direct limit under inclusion of cyclic groups C p N of order p N N 0, with the discrete topology.
2 Abstract and Applied Analysis When one talks of -extension, is variously considered as an indeterminate, a complex number C, or a p-adic number C p. If C p, then we normally assume 1 p <p 1/p 1, so that x expx log for x p 1. If C, then we assume that < 1. Let Z Z p lim, Z p N p a pz p, Z N 0<a<p 1.2 a p N Z p { x Z p x a mod p N}, where a Z lies in 0 a<p N. We use the following notation: Hence x 1 x 1. 1.3 limx x 1 1.4 for any x with x p 1 in the present p-adic case. The distribution μ a p N Z p is given as μ a p N Z p a [ p N ] 1.5 cf. 1 9. For the ordinary p-adic distribution μ 0 defined by μ 0 a p N Z p 1 p N, 1.6 we see lim 1 μ μ 0. 1.7 We say that f is a uniformly differentiable function at a point a Z p, we write f UDZ p, C p if the difference uotient F f x, y fx fy x y 1.8 has a limit f a as x, y a, a. Also we use the following notation: cf.1 5. x 1 x, 1.9 1
Min-Soo Kim et al. 3 In 1 3, Kim gave a detailed proof of fermionic p-adic -measures on Z p. He treated some interesting formulae-related -extension of Euler numbers and polynomials; and he defined fermionic p-adic -measures on Z p as follows: μ a p N a Z p [ ]. 1.10 p N By using the fermionic p-adic -measures, he defined the fermionic p-adic -integral on Z p as follows: p 1 I f fxdμ x lim [ ] fx x Z N p p N x0 1.11 for f UDZ p, C p cf. 1 3. Observe that p I 1 f limi f fxdμ 1 x lim fx 1 x. 1 Z N p x0 1.12 From 1.12,weobtain I 1 f1 I 1 f 2f0, 1.13 where f 1 x fx 1. By substituting fx e tx into 1.13, classical Euler numbers are defined by means of the following generating function: e tx 2 dμ 1 x Z p e t 1 t n E n n!. 1.14 These numbers are interpolated by the Euler zeta function which is defined as follows: ζ E s cf. 1 9. From1.12, we also obtain 1 n n1 n s, s C, 1.15 I f1 I f 2 f0, 1.16 where f 1 x fx 1. By substituting fx e tx into 1.13, -Euler numbers are defined by means of the following generating function: e tx dμ x 2 Z p e t 1 t n E n, n!. 1.17
4 Abstract and Applied Analysis These numbers are interpolated by the Euler -zeta function which is defined as follows: 1 n n ζ,e s 2 n s, s C, 1.18 n1 cf. 4. In 6, Ozden and Simsek defined generating function of -Euler numbers by 2 e tx 2 dμ x 1 Z p e t 1, 1.19 which are different from 1.17. But we observe that all these generating functions were obtained by the same fermionic p-adic -measures on Z p and the fermionic p-adic -integrals on Z p. In this paper, we define a multiple twisted Carlitz s type -zeta functions, which interpolated multiple twisted Carlitz s type -Euler polynomials at negative integers. This result gave us a partial answer of the problem proposed by Jang et al. 10, which is given by: Are there analytic multiple twisted Carlitz s type -zeta functions which interpolate multiple twisted Carlitz s type -Euler Bernoulli polynomials? 2. Preliminaries In 10, Jang and Ryoo defined -extension of Euler numbers and polynomials of higher order and studied multivariate -Euler zeta functions. They also derived sums of products of -Euler numbers and polynomials by using ferminonic p-adic -integral. In 5, 7, Ozden et al. defined multivariate Barnes-type Hurwitz -Euler zeta functions and l-functions. They also gave relation between multivariate Barnes-type Hurwitz -Euler zeta functions and multivariate -Euler l-functions. In this section, we consider twisted -extension of Euler numbers and polynomials of higher order and study multivariate twisted Barnes-type Hurwitz -Euler zeta functions and l-functions. Let UDZ h p, C p denote the space of all uniformly or strictly differentiable C p -valued functions on Z h p Z p Z p. For f UDZ h p, C p, the p-adic -integral on Z h p is defined by I h f f x 1,...,x h dμ x1 dμ xh Z p Z p 2.1 cf. 3. If 1, then 1 lim N [ ] p N h p x 1 0 p x h 0 f x 1,...,x h x 1 x h I h 1 f lim 1 Ih f lim p N x 1 0 p x h 0 f x 1,...,x h 1 x 1 x h. 2.2
Min-Soo Kim et al. 5 For a fixed positive integer d with d, p 1, we set For f UDZ h p, C p, Z X p lim N. 2.3 dp N Z I h 1 f f x 1,...,x h dμ 1 x1 dμ 1 xh, X p X p 2.4 cf. 2. We set fx 1,...,x h ω x 1 x h e xx 1 x h t in 2.2 and 2.4. Thenwehave Z p Z p ω x 1 x h e xx 1 x h t 2 dμ 1 x1 dμ 1 xh ωe t 1 2 ωe t 1 E h n,ωx tn n!, 2.5 where En,ωx h are the twisted Euler polynomials of order h. From 2.5,wenotethat ω x 1 x h ndμ 1 h x x1 x h x1 dμ 1 xh E n,ωx. Z p Z p 2.6 We give an application of the multivariate -deformed p-adic integral on Z h p in the fermionic sense related to 3.Let By substituting Z h p. 2.7 Z p Z p f x 1,...,x h ω x 1 x h e xx 1 x h t 2.8 into 2.1, we define twisted -extension of Euler numbers of higher order by means of the following generating function: Z h p ω x 1 x h e x 1 x h t 2 2 dμ x1 dμ xh ωe t 1 ωe t 1 E h t n n,,ω n!. 2.9
6 Abstract and Applied Analysis Then we have Z h p ω x 1 x h x1 x h ndμ x1 dμ xh E h n,,ω. 2.10 From 2.9,weobtain Z h p ω x 1 x h e xx 1 x h t 2 h dμ ext x1 dμ xh ωet 1 ωe t 1 En,,ωx h tn n!, 2.11 where En,,ωx h is called twisted -extension of Euler polynomials of higher order cf. 11.We note that if ω 1, then En,,ωx h En,x h and E h n,,ω E h n, cf. 6.Wealsonotethat E h n,,ωx n k0 n E h k k,,ω xn k. 2.12 The twisted -extension of Euler polynomials of higher order, E h n,,ωx, is defined by means of the following generating function: G h 2,ωx, t ωe t 1 2 e xt ωe t 1 2 h etx ω l 1 l 1 e l 1t ω l h l h e l ht 2 h l 1 0 En,,ωx h tn l h 0 ω l 1 l h l 1 l h e l 1 l h xt n!, 2.13 where t logω < π. From these generating functions of twisted -extension of Euler polynomials of higher order, we construct twisted multiple -Euler zeta functions as follows. For s C and x R with 0 <x 1, we define ζ h,ω,e s, x 2h ω l 1 l h l 1 l h l1 l h x s. 2.14 By the mth differentiation on both sides of 2.13 at t 0, we obtain the following Em,,ωx h d m G h,ωx, t dt 2 h ω l 1 l h l 1 l h x l 1 l h m t0 2.15 for m 0, 1,...
Min-Soo Kim et al. 7 From 2.14 and 2.15, we arrive at the following We set ζ h,ω,e m, x Eh m,,ωx, m 0, 1,... 2.16 X h p. 2.17 X p X p Let χ be Dirichlet s character with odd conductor d. We define twisted -extension of generalized Euler polynomials of higher order by means of the following generating function cf. 11: χ x 1 x h ω x 1 x h e xx 1 x h t dμ x1 dμ xh En,,ω,χx h tn n!. 2.18 X h p Note that En,,ω,χx h tn n! e xt X p X p e xt 1 d h N χ x 1 x h ω x 1 x h e x 1 x h t dμ x1 dμ xh 1 lim [ ] d 1 p N d p a 1 0 x 1 0 d 1 p a h 0 x h 0 ω a 1dx 1 a h dx h e a 1dx 1 a h dx h t a 1dx 1 a h dx h χ a 1 dx 1 a h dx h e xt 1 d 1 d h a 1,...,a h 0 χ a 1 a h ω a 1 a h a 1 a h e a 1 a h t 2.19 1 d 1 ω dpn dpn e dpn 1 d 1 ω dpn dpn e dpn lim lim N 1 dpn 1 ω d d e dt N 1 dpn 1 ω d d e dt h times e xt 1 d 1 d h a 1,...,a h 0 χ a 1 a h ω a 1 a h a 1 a h e a 1 a h t since 1 d 1 ω d d e 1 d dt 1 ω d d e }{{ dt } lim N pn 1 for 1 p < 1. 2.20
8 Abstract and Applied Analysis This allows us to rewrite 2.18 as En,,ω,χx h tn n! e xt 1 d 1 d h a 1,...,a h 0 1 d 1 ω d d e 1 d dt 1 ω d d e }{{ dt } χ a 1 a h ω a 1 a h a 1 a h e a 1 a h t 2 h ext d 1 a 1,...,a h 0 χ a 1 a h ω a 1 a h a 1 a h e a 1 a h t ω d d e dt x 1 ω d d e dt x h x 1 0 x h 0 } {{ } 2 h ext d 1 x 1,...,x h 0 a 1,...,a h 0 χ a 1 dx 1 a h dx h 2.21 ω a 1dx 1 a h dx h a 1dx 1 a h dx h e a 1dx 1 a h dx h t 2 h 1 l 1 l h χ l 1 l h ω l 1 l h l 1 l h e xl 1 l h t. By applying the mth derivative operator d/dt m t0 in the above euation, we have Em,,ω,χx h 2 h χ h l 1 l h 1 l i ω l i i l m x l1 l h 2.22 i1 for m 0, 1,... From these generating functions of twisted -extension of generalized Euler polynomials of higher order, we construct twisted multiple -Euler l-functions as follows. For s C and x R with 0 <x 1, we define l h,ω,e s, x, χ 2h χ l 1 l h h i1 1l i ω l i l i l1 l h x s. 2.23 From 2.22 and 2.23, we arrive at the following l h,ω,e m, x, χ Eh m,,ω,χx, m 0, 1,... 2.24
Min-Soo Kim et al. 9 Let s C and a i,f Z with F is an odd integer and 0 <a i <F,where i 1,...,h.Then twisted partial multiple -Euler ζ-functions are as follows: H h,ω,e s, a1,...,a h,x F 2 h l i a i mod F, i1,...,h 1 l 1 l h ω l 1 l h l 1 l h l1 l h x s. For i 1,...,h,substituting l i a i n i F with F is odd into 2.25,wehave s, a1,...,a h,x F H h,ω,e 2 h n 1,...,n h 0 2h a ω 1 a h 2 h F s 2 h F F a ω 1 a h 2 h F s F 2h 1 a 1n 1 F a h n h F ω a 1n 1 F a h n h F a 1n 1 F a h n h F a1 n 1 F a h n h F x s n 1,...,n h 0 ζ h F,ω F,E 1 n 1 n h ω F n 1 n h F n 1 n h n1 n h a 1 a h x /F s s, a 1 a h x F. 2.25 2.26 Then we obtain H h,ω,e s, a1,...,a h,x F 2h a ω 1 a h 2 h F s ζ h s, a 1 a h x. 2.27 F,ω F,E F F By using 2.12 and 2.27 and by substituting s m, m 0, 1,...,we get H h,ω,e m, a1,...,a h,x F 2h ω a 1 a h a1 a 2 h h x m F m k0 m F k E h. k a 1 a h x k, F,ω F Therefore, we modify twisted partial multiple -Euler zeta functions as follows: 2.28 H h,ω,e s, a1,...,a h,x F 2h ω a 1 a h a1 a 2 h h x s F k0 s F k E h. k a 1 a h x k, F,ω F Let χ be a Dirichlet character with conductors d and d F. From 2.23 and 2.27,wehave l h,ω,e s, x, χ 2h 2 h F F s F 1 ω a 1 a h a 1,...,a h 0 χ a 1 a h ζ h F,ω F,E F 1 a 1,...,a h 0 s, a 1 a h x F χ h a 1 a h H,ω,E s, x, a1,...,a h,x F. 2.29 2.30
10 Abstract and Applied Analysis 3. The multiple twisted Carlitz s type -Euler polynomials and -zeta functions Let us consider the multiple twisted Carlitz s type -Euler polynomials as follows: En,,ωx z,h [ x1 x h x ] n ωx 1 x h x 1z 1 x h z h dμ x1 dμ xh Z p Z p cf. 1, 3. These can be written as En,,ωx z,h 2h 1 n n i0 3.1 n ix 1 i 1 i 1 ω 1. 3.2 zi 1 ωzi h1 We may now mention the following formulae which are easy to prove: ω z E z,h n,,ωx 1E z,h n,,ωx 2 E z 1,h 1 n,,ω x. 3.3 From 3.2, we can derive generating function for the multiple twisted Carlitz s type -Euler polynomials as follows: En,,ωx z,h tn n! 2 h n n 1 n ix 1 i 1 i 1 ω 1 t n zi 1 ω zi h1 n! i0 2 h n n 1 n ix 1 i ω zi l 1 ω zi h1 l h t n i n! 2 h 1 n i0 1 l 1 l h l 1 0 n i0 l h 0 n xl 1 l h i 1 i i ω l 1 l h l 1zl 2 z 1 l h z h1 tn n! 2 h 1 l 1 l h ω l 1 l h l 1zl 2 z 1 l h z h1 2 h [ ] n t n x l1 l h n! ω l 1 l h l 1zl 2 z 1 l h z h1 e xl 1 l h t. Also, an obvious generating function for the multiple twisted Carlitz s type -Euler polynomials is obtained, from 3.2, by 3.4 n 1 j 2 h et/1 1 j jx 1 1 1 ω 1 zj Ez,h 1 ωzj h1 n,,ωx. j0 3.5
Min-Soo Kim et al. 11 From now on, we assume that C with < 1. From 3.2 and 3.4,wenotethat G z,h,ω x, t 2 h E z,h n,,ωx En,,ωx z,h tn 2h 1 n 2 h n i0 ω l 1 l h l 1zl 2 z 1 l h z h1 e xl 1 l h t n!, n ix 1 i 1 i 1 ω 1 zi 1 ω zi h1 ω l 1 l h l 1zl 2 z 1 l h z h1 [ x l 1 l h ] n. 3.6 3.7 Thus we can define the multiple twisted Carlitz s type -zeta functions as follows: ζ z,h,ω s, x 2 h ω l 1 l h l 1zl 2 z 1 l h z h1 [ ] s. 3.8 x l1 l h In 12, Proposition 3, Yamasaki showed that the series ζ z,h,ω s, x converges absolutely for Re z >h 1, and it can be analytically continued to the whole complex plane C. Note that if h 1, then ζ z,h,ω s, x ζ,ωs, z ω l lz x 2 x l s. 3.9 l0 In 13, Wakayama and Yamasaki studied -analogue of the Hurwitz zeta function ζs, x 1 n x s defined by the -series with two complex variable s, z C: ζ z s, x 3.10 nxz x n s, Rez > 0, 3.11 and special values at nonpositive integers of the -analogue of the Hurwitz zeta function. Therefore, by the mth differentiation on both sides of 3.6 at t 0, we obtain the following: Em,,ωx z,h d m G z,h,ω x, t dt t0 2 h ω l 1 l h l 1zl 2 z 1 l h z h1 [ 3.12 ] m x l 1 l h for m 0, 1,...
12 Abstract and Applied Analysis From 3.7, 3.8, and3.12, wehave3.13 which shows that the multiple twisted Carlitz s type -zeta functions interpolate the multiple twisted Carlitz s type -Euler numbers and polynomials. For m 0, 1,...,we have ζ z,h,ω m, x E z,h m,,ωx, 3.13 where x R and 0 <x 1. Thus, we derive the analytic multiple twisted Carlitz s type -zeta functions which interpolate multiple twisted Carlitz s type -Euler polynomials. This gives a part of the answer to the uestion proposed in 10. 4. Remarks For nonnegative integers m and n, we define the -binomial coefficient [ m n ] by [ ] m ; m, n ; n ; m n where a; m m 1 k0 1 ak for m 1anda; 0 1. For h N, it holds that l 1,...,l h 0 l 1 l h l l 12l 2 hl h lh [ l h 1 h 1 ] 4.1 4.2 cf. 12, Lamma 2.3. From3.8, it is easy to see that ζ z,h,ω s, x 2 h l0 l 1,...,l h 0 l 1 l h l ω l 1 l h z1l 1 l h l 1 2l 2 hl h [ x l1 l h ] s 2 h ω l z1l l x s 2 h l0 l0 [ l h 1 h 1 ] l 1,...,l h 0 l 1 l h l l 12l 2 hl h ω l z h1l l x s. 4.3 We set m! m m 1 1 for m N. The following identity has been studied in 12: [ ] l h 1 1 h 1 l x lj h 1 x j h 1 h 1! lh 1 k P k,h xl xk, 4.4 j1 k0 where P k x, 0 k h 1, is a function of x defined by,h P k,hx 1h 1 k h 1! 1 m 1 < <m h 1 k h 1 m 1 m h 1 k [ ] [x x m1 m h 1 k ] 4.5
Min-Soo Kim et al. 13 for 0 k h 2andP h 1,h x 1/h 1!. By using 3.9, 4.3,and4.5,wehave and so,ω s, x 2 h h 1 P k,h x ω l z kl k0 l0 l x s k ζ z,h ζ z,h,ω m, x 2 h 1 h 1 k0 2 h 1 h 1 k0 P k,h xζz k P k,h xζz k,ω s k, x, 4.6,ω m k, x. 4.7 The values of ζ z,h,ω m, x at h 2, 3 are given explicitly as follows: ζ z,2,ω m, x 1 ζ z 1,ω m 1,x x 1 ζ,ω m, z x, Acknowledgment { ζ z,3,ω m, x 1 ζ z 2,ω m 2,x x 1 2 z 1 x 2 ζ,ω m 1,x } 3 x 1 x 2 ζ,ω m, z x. This work is supported by Kyungnam University Foundation Grant no. 2007. References 1 T. Kim, On the -extension of Euler and Genocchi numbers, Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458 1465, 2007. 2 T. Kim, On p-adic interpolating function for -Euler numbers and its derivatives, Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598 608, 2008. 3 T. Kim, On the multiple -Genocchi and Euler numbers, to appear in Russian Journal of Mathematical Physics. 4 T. Kim, M.-S. Kim, L.-C. Jang, and S.-H. Rim, New -Euler numbers and polynomials associated with p-adic -integrals, Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 243 252, 2007. 5 H. Ozden, I. N. Cangul, and Y. Simsek, Multivariate interpolation functions of higher-order -Euler numbers and thier applications, Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008. 6 H. Ozden and Y. Simsek, Interpolation function of the h, -extension of twisted Euler numbers, Computers & Mathematics with Applications. In press. 7 H. Ozden, I. N. Cangul, and Y. Simsek, Remarks on sum of products of h, -twisted Euler polynomials and numbers, Journal of Ineualities and Applications, vol. 2008,Article ID816129,8 pages, 2008. 8 T. Kim, L.-C. Jang, and C.-S. Ryoo, Note on -extensions of Euler numbers and polynomials of higher order, Journal of Ineualities and Applications, vol. 2008, Article ID 371295, 9 pages, 2008. 9 Y. Simsek, -analogue of twisted l-series and -twisted Euler numbers, Journal of Number Theory,vol. 110, no. 2, pp. 267 278, 2005. 10 L.-C. Jang and C.-S. Ryoo, A note on the multiple twisted Carlitz s type -Bernoulli polynomials, Abstract and Applied Analysis, vol. 2008, Article ID 498173, 7 pages, 2008. 11 L.-C. Jang, Multiple twisted -Euler numbers and polynomials associated with p-adic -integrals, Advances in Difference Euations, vol. 2008, Article ID 738603, 11 pages, 2008. 12 Y. Yamasaki, On -analogues of the Barnes multiple zeta functions, Tokyo Journal of Mathematics,vol. 29, no. 2, pp. 413 427, 2006. 13 M. Wakayama and Y. Yamasaki, Integral representations of -analogues of the Hurwitz zeta function, Monatshefte für Mathematik, vol. 149, no. 2, pp. 141 154, 2006. 4.8