Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 640152, 7 pages doi:10.1155/2009/640152 ResearchArticleOntheIdentitiesofSymmetryfortheGeneralizedBernoulliPolynomialsAttachedtoχofHigher Order Taekyun Kim, 1 Lee-Chae Jang, 2 Young-Hee Kim, 1 and Kyung-Won Hwang 3 1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea 2 Department of Mathematics and Computer Science, Konkook University, Chungju 139-701, South Korea 3 Department of General Education, Kookmin University, Seoul 136-702, South Korea Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr Received 5 June 2009; Accepted 5 August 2009 Recommended by Vijay Gupta We give some interesting relationships between the power sums and thegeneralizedBernoulli numbers attachedtoχofhigher order using multivariate p-adic invariant integral on Z p . 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, 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, and Z N ∪{0}.Letν p be the normalized exponential valuation of C p with |p| p p −ν p p p −1 see 1–24. Let UDZ p be the space of uniformly differentiable function on Z p .Letd be a fixed positive integer. For n ∈ N,let 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 , 1.1 2 Journal of Inequalities and Applications where a ∈ Z lies in 0 ≤ a<dp N . For f ∈ UDX,thep-adic invariant integral on X is defined as I f X f x dx lim N →∞ 1 dp N dp N −1 x0 f x 1.2 see 11–19.From1.2,wenotethat I f 1 I f f 0 , 1.3 where f 0df x/dx| x0 and f 1 xfx 1.Letf n xfx nn ∈ N. Then we can derive the following equation from 1.3: I f n I f n−1 i0 f i 1.4 see 1–11.Letχ be the Dirichlet’s character with conductor d ∈ N. Then thegeneralizedBernoullipolynomialsattachedtoχ are defined as d−1 a0 χ a e at t e dt − 1 e xt ∞ n0 B n,χ x t n n! , 1.5 and thegeneralizedBernoulli numbers attachedto χ, B n,χ , are defined as B n,χ B n,χ 0 see 1–20, 25. The purpose of this paper is to derive some identitiesofsymmetryforthegeneralizedBernoullipolynomialsattachedtoχofhigher order. 2. Symmetric Properties fortheGeneralizedBernoulliPolynomialsofHigher Order Let χ be the Dirichlet’s character with conductor d ∈ N. Then we note that X χ x e xt dx t d−1 i0 χ i e it e dt − 1 ∞ n0 B n,χ t n n! , 2.1 where B n,χ are the nth generalizedBernoulli numbers attachedtoχ see 7, 9, 15, 25.Now we also see that thegeneralizedBernoullipolynomialsattachedtoχ are given by X χ y e xyt dy t d−1 i0 χ i e it e dt − 1 e xt ∞ n0 B n,χ x t n n! . 2.2 By 2.1 and 2.2, we have X χ x x n dx B n,χ 2.3 Journal of Inequalities and Applications 3 see 15, 25 ,and X χ y x y n dy B n,χ x 2.4 see 1–19, 25 . For n ∈ N,weobtainthat X f x n dx X f x dx n−1 i0 f i , 2.5 where f idf x/dx| xi . Thus, we have 1 t X χ x e ndx t dx − X χ x e xt dx nd X χ x e xt dx X e ndxt dx e ndt − 1 e dt − 1 d−1 i0 χ i e it . 2.6 Then 1 t X χ x e ndx t dx − X χ x e xt dx nd−1 l0 χ l e lt ∞ k0 nd−1 l0 χ l l k t k k! . 2.7 Let us define the p-adic function T k χ, n as follows: T k χ, n n l0 χ l l k 2.8 see 25.By2.7 and 2.8,weseethat 1 t X χ x e ndx t dx − X χ x e xt dx ∞ k0 T k χ, nd − 1 t k k! 2.9 see 25. T hus, we have X χ x nd x k dx − X χ x x k dx kT k−1 χ, nd − 1 , k,n,d∈ N. 2.10 This means that B k,χ nd − B k,χ kT k−1 χ, nd − 1 , k,n,d∈ N 2.11 see 25. 4 Journal of Inequalities and Applications ThegeneralizedBernoullipolynomialsattachedtoχof order k, which is denoted by B k n,χ x, are defined as t d−1 i0 χie it e dt − 1 k e xt ∞ n0 B k n,χ x t n n! . 2.12 Then the values of B k n,χ x at x 0 are called thegeneralizedBernoulli numbers attachedtoχof order k. When k 1, thepolynomialsof numbers are called thegeneralizedBernoullipolynomials or numbers attachedto χ.Letw 1 ,w 2 ∈ N. Then we set K m, χ; w 1 ,w 2 d X m m i1 χ x i e m i1 x i w 2 xw 1 t m i1 dx i X m m i1 χ x i e m i1 x i w 1 yw 2 t m i1 dx i X e dw 1 w 2 xt dx , 2.13 where X m f x 1 , ,x m dx 1 ···dx m X ··· X f x 1 , ,x m dx 1 ···dx m . 2.14 In 2.13,wenotethatKm, χ; w 1 ,w 2 is symmetric in w 1 ,w 2 .From2.13, we derive K m, χ; w 1 ,w 2 X m m i1 χ x i e m i1 x i w 1 t dx 1 ···dx m e w 1 w 2 xt d X χ x m e w 2 x m t dx m X e dw 1 w 2 xt dx × X m−1 m−1 i1 χ x i e m−1 i1 x i w 2 t dx 1 ···dx m−1 e w 1 w 2 yt . 2.15 It is easy to see that w 1 d X χ x e xt dx X e dw 1 xt dx ∞ k0 w 1 d−1 i0 χ i i k t k k! ∞ k0 T k χ, w 1 d − 1 t k k! , e w 1 w 2 xt X m m i1 χ x i e m i1 x i w 1 t dx 1 ···dx m e w 1 w 2 xt w 1 t e dw 1 t − 1 d−1 a0 χ a e w 1 at m ∞ n0 B m n,χ w 2 x w 1 n t n n! . 2.16 Journal of Inequalities and Applications 5 From 2.16,wenotethat K m, χ; w 1 ,w 2 ∞ l0 B m l,χ w 2 x w 1 l t l l! ∞ k0 T k χ, w 1 d − 1 w 2 k t k k! ∞ i0 B m−1 i,χ w 1 y w 2 i t i i! 1 w 1 ∞ n0 ⎡ ⎣ n j0 n j w j 2 w n−j−1 1 B m n−j,χ w 2 x j k0 T k χ, w 1 d − 1 j k B m−1 j−k,χ w 1 y ⎤ ⎦ t n n! . 2.17 By thesymmetryof Km, χ; w 1 ,w 2 in w 1 and w 2 ,weseethat K m, χ; w 1 ,w 2 ∞ n0 ⎡ ⎣ n j0 n j w j 1 w n−j−1 2 B m n−j,χ w 1 x j k0 T k χ, w 2 d − 1 j k B m−1 j−k,χ w 2 y ⎤ ⎦ t n n! . 2.18 By comparing the coefficients onthe both sides of 2.17 and 2.18, we see the following theorem. Theorem 2.1. For d, w 1 ,w 2 ∈ N,n≥ 0,m≥ 1, one has n j0 n j w j 2 w n−j−1 1 B m n−j,χ w 2 x j k0 T k χ, w 1 d − 1 j k B m−1 j−k,χ w 1 y n j0 n j w j 1 w n−j−1 2 B m n−j,χ w 1 x j k0 T k χ, w 2 d − 1 j k B m−1 j−k,χ w 2 y . 2.19 Remark 2.2. Let y 0andm 1in1.4. Then we have n j0 n j w j 2 w n−j−1 1 B n−j,χ w 2 x T j χ, w 1 d − 1 n j0 n j w j 1 w n−j−1 2 B n−j,χ w 1 x T j χ, w 2 d − 1 2.20 see 25. 6 Journal of Inequalities and Applications We also calculate that K m, χ; w 1 ,w 2 ∞ n0 n k0 n k w k−1 1 w n−k 2 B m−1 n−k,χ w 1 y dw 1 −1 i0 B m k,χ w 2 x w 2 w 1 i t n n! . 2.21 From the symmetric property of Km, χ; w 1 ,w 2 in w 1 and w 2 , we derive K m, χ; w 1 ,w 2 ∞ n0 n k0 n k w k−1 2 w n−k 1 B m−1 n−k,χ w 2 y dw 2 −1 i0 B m k,χ w 1 x w 1 w 2 i t n n! . 2.22 By comparing the coefficients onthe both sides of 2.21 and 2.22, we obtain the following theorem. Theorem 2.3. For w 1 ,w 2 ∈ N,n∈ Z,m∈ N, one has n k0 n k w k−1 1 w n−k 2 B m−1 n−k,χ w 1 y dw 1 −1 i0 B m k,χ w 2 x w 2 w 1 i n k0 n k w k−1 2 w n−k 1 B m−1 n−k,χ w 2 y dw 2 −1 i0 B m k,χ w 1 x w 1 w 2 i . 2.23 Remark 2.4. Let y 0andm 1in2.23. We have w n−1 1 dw 1 −1 i0 B n,χ w 2 x w 2 w 1 i w n−1 2 dw 2 −1 i0 B n,χ w 1 x w 1 w 2 i 2.24 see 25. Acknowledgment The present research has been conducted by theresearch Grant ofthe Kwangwoon University in 2009. Journal of Inequalities and Applications 7 References 1 M. Cenkci, “The p-adic generalized twisted h, q-Euler-l-function and its applications,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 37–47, 2007. 2 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. 3 L C. Jang, S D. Kim, D W. Park, and Y S. 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The purpose of this paper is to derive some identities of symmetry for the generalized Bernoulli polynomials attached to χ of higher order. 2 Corporation Journal of Inequalities and Applications Volume 2009, Article ID 640152, 7 pages doi:10.1155/2009/640152 Research Article On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached. numbers attached to χ of order k. When k 1, the polynomials of numbers are called the generalized Bernoulli polynomials or numbers attached to χ. Letw 1 ,w 2 ∈ N. Then we set K m, χ; w 1 ,w 2 d X m m i1 χ x i e m i1 x i w 2 xw 1 t m i1 dx i X m m i1 χ x i e m i1 x i w 1 yw 2 t m i1 dx i X e dw 1 w 2 xt dx , 2.13 where X m f x 1 ,