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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 826548, 8 pages doi:10.1155/2010/826548 Research Article On the Symmetric Properties of the Multivariate p-Adic Invariant Integral on Z p Associated with the Twisted Generalized Euler Polynomials of H igher Order Taekyun Kim, 1 Byungje Lee, 2 and Young-Hee Kim 1 1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea 2 Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea Correspondence should be addressed to Young-Hee Kim, yhkim@kw.ac.kr Received 6 November 2009; Revised 11 March 2010; Accepted 14 March 2010 Academic Editor: Ulrich Abel Copyright q 2010 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. We study the symmetric properties for the multivariate p-adic invariant integral on Z p related to the twisted generalized Euler polynomials of higher order. 1. Introduction Let p be a fixed prime number. Throughout this paper, the symbols Z, Z p , Q p ,andC p denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively. Let N be the set of natural numbers. The normalized valuation in C p is denoted by |·| p with |p| p 1/p.LetUDZ p be the space of uniformly differentiable function on Z p . For f ∈ UDZ p , the fermionic p-adic invariant integral on Z p is defined as I f Z p f x dμ x lim N →∞ p N −1 x0 f x μ x p N Z p lim N →∞ p N −1 x0 f x −1 x 1.1 see 1–25. For n ∈ N,wenotethat Z p f x n dμ x −1 n Z p f x dμ x n−1 l0 −1 n−1−l f l see 5 . 1.2 2 Journal of Inequalities and Applications Let d be a fixed odd positive integer. For N ∈ N,weset X X d lim ←− N Z/dp N Z,X 1 Z p , X ∗ 0<a<dp a,p1 a dpZ p , a dp N Z p x ∈ X | x ≡ a mod dp N , 1.3 where a ∈ Z lies in 0 ≤ a<dp N see 1–13. It is well known that for f ∈ UDZ p , X f x dμ x Z p f x dμ x . 1.4 For n ∈ N,letC p n be the cyclic group of order p n .Thatis,C p n {ξ | ξ p n 1}.Thep-adic locally constant space, T p , is defined by T p lim n →∞ C p n n≥1 C p n . Let χ be Dirichlet’s character with conductor d ∈ N and let ξ ∈ T p . Then the generalized twisted Bernoulli polynomials B n,χ,ξ x attached to χ are defined as t d−1 a0 χ a ξ a e at ξ d e dt − 1 e xt ∞ n0 B n,χ,ξ x t n n! see 10 . 1.5 In 4, 7, 10–12, t he generalized twisted Bernoulli polynomials of order k attached to χ are also defined as follows: t d−1 a0 χ a ξ a e at ξ d e dt − 1 ×···× t d−1 a0 χ a ξ a e at ξ d e dt − 1 k-times e xt ∞ n0 B k n,χ,ξ x t n n! . 1.6 Recently, the symmetry identities for the generalized twisted Bernoulli polynomials and the generalized twisted Bernoulli polynomials of order k are studied in 4, 12. In this paper, we study the symmetric properties of the multivariate p-adic invariant integral on Z p . From these symmetric properties, we derive the symmetry identities for the twisted generalized Euler polynomials of higher order. In 14, Kim gave the relation between the power sum polynomials and the generalized higher-order Euler polynomials. The main purpose of this paper is to give the symmetry identities for the twisted generalized Euler polynomials of higher order using the symmetric properties of the multivariate p-adic invariant integral on Z p . Journal of Inequalities and Applications 3 2. Symmetry Identities for the Twisted Generalized Euler Polynomials of Higher Order Let χ be Dirichlet’s character with an odd conductor d ∈ N.Thatis,d ∈ N with d ≡ 1 mod 2. For ξ ∈ T p , the twisted generalized Euler polynomials attached to χ, E n,χ,ξ x, are defined as X χ y ξ y e xyt dμ y 2 d−1 a0 −1 a χ a ξ a e at ξ d e dt 1 e xt ∞ n0 E n,χ,ξ x t n n! see 12 . 2.1 In the special case x 0, E n,χ,ξ E n,χ,ξ 0 are called the nth twisted generalized Euler numbers attached to χ. From 2.1,wenotethat X χ y ξ y x y m dμ y E m,χ,ξ x ,m∈ N ∪ { 0 } . 2.2 For n ∈ N with n ≡ 1 mod 2, we have X χ x ξ x e xndt dμ x X χ x ξ x e xt dμ x 2 nd−1 l0 −1 l χ l ξ l e lt . 2.3 Let T k,χ,ξ n n l0 −1 l χlξ l l k . Then we see that ξ nd X χ x ξ x e xndt dμ x X χ x ξ x e xt dμ x 2 X e xt χ x ξ x dμ x X e ndxt ξ ndx dμ x 2 ∞ k0 T k,χ,ξ nd − 1 t k k! . 2.4 Now we define the twisted generalized Euler polynomials E k n,χ,ξ x of order k attached to χ as follows: e xt 2 d−1 a0 −1 a χaξ a e at ξ d e dt 1 k ∞ n0 E k n,χ,ξ x t n n! . 2.5 In the special case x 0, E k n,χ,ξ E k n,χ,ξ 0 are called the nth twisted generalized Euler numbers of order k. 4 Journal of Inequalities and Applications Let w 1 ,w 2 ,d ∈ N with w 1 ≡ 1,w 2 ≡ 1, and d ≡ 1 mod 2. Then we set J m χ,ξ w 1 ,w 2 | x X m m i1 χ x i ξ m i1 x i w 1 e m i1 x i w 2 xw 1 t dμ x 1 ···dμ x m X ξ dw 1 w 2 x e dw 1 w 2 xt dμ x × X m m i1 χ x i ξ m i1 x i w 2 e m i1 x i w 1 yw 2 t dμ x 1 ···dμ x m , 2.6 where X m f x 1 , ,x m dμ x 1 ···dμ x m X ··· X m-times f x 1 , ,x m dμ x 1 ···dμ x m . 2.7 From 2.6,wenotethat J m χ,ξ w 1 ,w 2 | x X m m i1 χ x i ξ m i1 x i w 1 e m i1 x i w 1 t dμ x 1 ···dμ x m e w 1 w 2 xt × X χ x m ξ w 2 x m e w 2 x m t dμ x m X ξ dw 1 w 2 x e dw 1 w 2 xt dμ x e w 1 w 2 yt × X m−1 m−1 i1 χ x i ξ m−1 i1 x i w 2 e m−1 i1 x i w 2 t dμ x 1 ···dμ x m−1 . 2.8 From 2.4, we can easily derive the following equation: X χ x ξ x e xt dμ x X ξ dw 1 x e dw 1 xt dμ x dw 1 −1 l0 −1 l χ l ξ l e lt ∞ k0 T k,χ,ξ dw 1 − 1 t k k! . 2.9 It is not difficult to show that e w 1 w 2 xt X m m i1 χ x i ξ m i1 x i w 1 e m i1 x i w 1 t dμ x 1 ···dμ x m 2 d−1 a0 −1 a χaξ aw 1 e aw 1 t ξ dw 1 e dw 1 t 1 m e w 1 w 2 xt ∞ k0 E m k,χ,ξ w 1 w 2 x w k 1 t k k! . 2.10 Journal of Inequalities and Applications 5 By 2.8, 2.9,and2.10,weseethat J m χ,ξ w 1 ,w 2 | x ∞ l0 E m l,χ,ξ w 1 w 2 x w l 1 t l l! ∞ k0 T k,χ,ξ w 2 w 1 d − 1 w k 2 t k k! × ∞ i0 E m−1 i,χ,ξ w 2 w 1 y w i 2 t i i! ∞ n0 ⎛ ⎝ n j0 n j w j 2 w n−j 1 E m n−j,χ,ξ w 1 w 2 x × j k0 j k T k,χ,ξ w 2 w 1 d − 1 E m−1 j−k,χ,ξ w 2 w 1 y t n n! . 2.11 In the viewpoint of the symmetry of J m χ,ξ w 1 ,w 2 | x for w 1 and w 2 , we have J m χ,ξ w 1 ,w 2 | x ∞ n0 ⎛ ⎝ n j0 n j w j 1 w n−j 2 E m n−j,χ,ξ w 2 w 1 x × j k0 j k T k,χ,ξ w 1 w 2 d − 1 E m−1 j−k,χ,ξ w 1 w 2 y t n n! . 2.12 Comparing the coefficients on both sides of 2.11 and 2.12, we obtain the following theorem. Theorem 2.1. Let w 1 ,w 2 ,d ∈ N with w 1 ≡ 1,w 2 ≡ 1, and d ≡ 1 mod 2. For n ∈ N ∪{0} and m ∈ N, one has n j0 n j w j 2 w n−j 1 E m n−j,χ,ξ w 1 w 2 x j k0 j k T k,χ,ξ w 2 w 1 d − 1 E m−1 j−k,χ,ξ w 2 w 1 y n j0 n j w j 1 w n−j 2 E m n−j,χ,ξ w 2 w 1 x j k0 j k T k,χ,ξ w 1 w 2 d − 1 E m−1 j−k,χ,ξ w 1 w 2 y . 2.13 Let m 1andy 0inTheorem 2.1. Then we also have the following corollary. Corollary 2.2. For w 1 ,w 2 ,d ∈ N with w 1 ≡ 1,w 2 ≡ 1, and d ≡ 1 mod 2, one has n m0 n m E m,χ,ξ w 1 w 2 x w m 1 w n−m 2 T n−m,χ,ξ w 2 w 1 d − 1 n m0 n m E m,χ,ξ w 2 w 1 x w n−m 1 w m 2 T n−m,χ,ξ w 1 w 2 d − 1 see 2 . 2.14 6 Journal of Inequalities and Applications Let χ be the trivial character and d 1. Then we also have the following corollary. Corollary 2.3. Let w 1 ,w 2 ∈ N with w 1 ≡ 1,w 2 ≡ 1 mod 2. Then one has n j0 n j w n−j 1 w j 2 E n−j,ξ w 1 w 2 x T k,ξ w 2 w 1 − 1 n j0 n j w j 1 w n−j 2 E n−j,ξ w 2 w 1 x T k,ξ w 1 w 2 − 1 , 2.15 where E n,ξ x are the nth twisted Euler polynomials. If we take w 2 1inCorollary 2.3, then we obtain the following corollary. Corollary 2.4 Distribution for the twisted Euler polynomials. For w 1 ∈ N with w 1 ≡ 1 mod 2, one has E n,ξ x n i0 n i w i 1 E i,ξ w 1 x T n−i,ξ w 1 − 1 . 2.16 From 2.6, we can derive that J m χ,ξ w 1 ,w 2 | x w 1 d−1 l0 χ l −1 l ξ w 2 l X m m i1 χ x i ×ξ m i1 x i w 1 e w 1 m i1 x i w 2 /w 1 lw 2 xt dμ x 1 ···dμ x m × X m−1 m−1 i1 χ x i ξ m−1 i1 x i w 2 e m−1 i1 x i w 2 t dμ x 1 ···dμ x m−1 ∞ n0 n k0 n k w k 1 w n−k 2 E m−1 n−k,χ,ξ w 2 w 1 y × w 1 d−1 l0 χ l −1 l ξ w 2 l E m k,χ,ξ w 1 w 2 x w 2 w 1 l t n n! . 2.17 By the symmetry property of J m χ,ξ w 1 ,w 2 | x in w 1 and w 2 ,wealsoseethat J m χ,ξ w 1 ,w 2 | x ∞ n0 n k0 n k w k 2 w n−k 1 E m−1 n−k,χ,ξ w 1 w 2 y w 2 d−1 l0 χ l −1 l ξ w 1 l E m k,χ,ξ w 2 w 1 x w 1 w 2 l t n n! . 2.18 Journal of Inequalities and Applications 7 Comparing the coefficients on both sides of 2.17 and 2.18, we obtain the following theorem which shows t he relationship between the power sums and the twisted generalized Euler polynomials of higher order. Theorem 2.5. Let w 1 ,w 2 ,d ∈ N with w 1 ≡ 1,w 2 ≡ 1, and d ≡ 1 mod 2. For n ∈ N ∪{0} and m ∈ N, one has n k0 n k w k 1 w n−k 2 E m−1 n−k,χ,ξ w 2 w 1 y w 1 d−1 l0 χ l −1 l ξ w 2 l E m k,χ,ξ w 1 w 2 x w 2 w 1 l n k0 n k w k 2 w n−k 1 E m−1 n−k,χ,ξ w 1 w 2 y w 2 d−1 l0 χ l −1 l ξ w 1 l E m k,χ,ξ w 2 w 1 x w 1 w 2 l . 2.19 If we take x 0, y 0, and m 1inTheorem 2.5, then we have the following identity: n k0 n k E k,χ,ξ w 1 w k 1 w n−k 2 T n−k,χ,ξ w 2 w 1 d − 1 n k0 n k E k,χ,ξ w 2 w k 2 w n−k 1 T n−k,χ,ξ w 1 w 2 d − 1 . 2.20 Acknowledgment The present research has been conducted by the research grant of Kwangwoon University in 2010. References 1 T. Kim and Y H. Kim, “On the symmetric properties for the generalized twisted Bernoulli polynomials,” Journal of Inequalities and Applications, vol. 2009, Article ID 164743, 9 pages, 2009. 2 T. Kim, “Symmetry identities for the twisted generalized Euler polynomials,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 2, pp. 151–155, 2009. 3 T. Kim, K. H. Park, and K W. Hwang, “On the identities of symmetry for the ζ-Euler polynomials of higher order,” Advances in Difference Equations, vol. 2009, Article ID 273545, 9 pages, 2009. 4 S H. Rim, Y H. Kim, B. Lee, and T. Kim, “Some identities of the generalized twisted Bernoulli numbers and polynomials of higher order,” Journal of Computational Analysis and Applications, vol. 12, no. 3, pp. 695–702, 2010. 5 T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on Z p ,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009. 6 M. Cenkci, Y. Simsek, and V. Kurt, “Multiple two-variable p-adic q-L-function and its behavior at s 0,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 447–459, 2008. 7 Y. Simsek, “On p-adic twisted -L-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006. 8 T. Kim and C. Adiga, “Sums of products of generalized Bernoulli numbers,” International Mathematical Journal, vol. 5, no. 1, pp. 1–7, 2004. 9 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order w-q-Genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009. 10 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on q-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009. 8 Journal of Inequalities and Applications 11 M. Cenkci, “The p-adic generalized twisted h, q-ζ-l-function and its applicaitons,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 37–47, 2007. 12 K. H. Park, “On interpolation functions of the generalized twisted h, q-Euler polynomials,” Journal of Inequalities and Applications, vol. 2009, Article ID 946569, 17 pages, 2009. 13 S H. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008. 14 T. Kim, “Symmetry properties of the generalized higher-order Euler polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 2, pp. 1–5, 2010. 15 T. Kim, “Note on the Euler q-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009. 16 T. Kim, “On a q-analogue of the p-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999. 17 T. Kim, “q-generalized Euler numbers and polynomials,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 293–298, 2006. 18 T. Kim, “Multiple p-adic L-function,” Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151– 157, 2006. 19 H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241–268, 2005. 20 T. Kim, “Power series and asymptotic series associated with the q -analog of the two-variable p-adic L-function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005. 21 T. Kim, “Analytic continuation of multiple q-zeta functions and their values at negative integers,” Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71–76, 2004. 22 T. Kim, “Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials,” Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91–98, 2003. 23 T. Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003. 24 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. 25 T. Kim, “Note on Dedekind type DC sums,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 249–260, 2009. . symmetric properties of the multivariate p-adic invariant integral on Z p . From these symmetric properties, we derive the symmetry identities for the twisted generalized Euler polynomials of higher. cited. We study the symmetric properties for the multivariate p-adic invariant integral on Z p related to the twisted generalized Euler polynomials of higher order. 1. Introduction Let p be a fixed. generalized Euler polynomials of higher order using the symmetric properties of the multivariate p-adic invariant integral on Z p . Journal of Inequalities and Applications 3 2. Symmetry Identities for the
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