Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 164743, 8 pages doi:10.1155/2009/164743 ResearchArticleOntheSymmetricPropertiesfortheGeneralizedTwistedBernoulli Polynomials Taekyun Kim and Young-Hee Kim Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea Correspondence should be addressed to Young-Hee Kim, yhkim@kw.ac.kr Received 6 July 2009; Accepted 18 October 2009 Recommended by Narendra Kumar Govil We study the symmetry forthegeneralizedtwistedBernoulli polynomials and numbers. We give some interesting identities of the power sums and thegeneralizedtwistedBernoulli polynomials using thesymmetricpropertiesforthe p-adic invariant integral. Copyright q 2009 T. Kim and Y H. Kim. 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 . Let UDZ p be the space of uniformly differentiable function on Z p . For f ∈ UDZ p , the p-adic invariant integral on Z p is defined as I f Z p f x dx lim N →∞ 1 p N p N −1 x0 f x 1.1 see 1.From1.1,wenotethat I f 1 I f f 0 , 1.2 2 Journal of Inequalities and Applications where f 0df x/dx| x0 and f 1 xfx 1. For n ∈ N,letf n xfx n. Then we can derive the following equation from 1.2: I f n I f n−1 i0 f i 1.3 see 1–7. Let d 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 dp Z p , a dp N Z p x ∈ X | x ≡ a mod dp N , 1.4 where a ∈ Z lies in 0 ≤ a<dp N . It is easy to see that X f x dx Z p f x dx, for f ∈ UD Z p . 1.5 The ordinary Bernoulli polynomials B n x are defined as t e t − 1 e xt ∞ n0 B n x t n n! , 1.6 and theBernoulli numbers B n are defined as B n B n 0see 1–19. For n ∈ N,letT p be the p-adic locally constant space defined by T p n≥1 C p n lim n →∞ C p n , 1.7 where C p n {ω | ω p n 1} is the cyclic group of order p n . It is well known that thetwistedBernoulli polynomials are defined as t ξe t − 1 e xt ∞ n0 B n,ξ x t n n! ,ξ∈ T p , 1.8 and thetwistedBernoulli numbers B n,ξ are defined as B n,ξ B n,ξ 0see 15–18. Let χ be Dirichlet’s character with conductor d ∈ N. Then thegeneralizedtwistedBernoulli polynomials B n,χ,ξ x attached to χ are defined as follows: d−1 a0 χ a ξ a e at t ξ d e dt − 1 e xt ∞ n0 B n,χ,ξ x t n n! ,ξ∈ T p . 1.9 Journal of Inequalities and Applications 3 ThegeneralizedtwistedBernoulli numbers attached to χ, B n,χ,ξ , are defined as B n,χ,ξ B n,χ,ξ 0see 16. Recently, many authors have studied thesymmetricproperties of the p-adic invariant integrals on Z p , which gave some interesting identities fortheBernoulli and the Euler polynomials cf. 3, 6, 7, 13, 14, 20–27. The authors of this paper have established various identities by thesymmetricproperties of the p-adic invariant integrals and investigated interesting relationships between the power sums and theBernoulli polynomials see 2, 3, 6, 7, 13. ThetwistedBernoulli polynomials and numbers and thetwisted Euler polynomials and numbers are very important in several fields of mathematics and physicscf. 15–18. The second author has been interested in thetwisted Euler numbers and polynomials and thetwistedBernoulli polynomials and studied the symmetry of power sum and twistedBernoulli polynomials see 11–13. The purpose of this paper is to study the symmetry forthegeneralizedtwistedBernoulli polynomials and numbers attached to χ.InSection 2, we give interesting identities forthe power sums and thegeneralizedtwistedBernoulli polynomials using thesymmetricpropertiesforthe p-adic invariant integral. 2. Symmetry fortheGeneralizedTwistedBernoulli Polynomials Let χ be Dirichlet’s character with conductor d ∈ N. For ξ ∈ T p , we have X χ x ξ x e xt dx t d−1 i0 χ i ξ i e it ξ d e dt − 1 ∞ n0 B n,χ,ξ t n n! , 2.1 where B n,χ,ξ are the nth generalizedtwistedBernoulli numbers attached to χ.Wealsoseethat thegeneralizedtwistedBernoulli polynomials attached to χ are given by X χ y ξ y e xyt dy t d−1 i0 χ i ξ i e it ξ d e dt − 1 e xt ∞ n0 B n,χ,ξ x t n n! . 2.2 By 2.1 and 2.2,weseethat X χ x ξ x x n dx B n,χ,ξ , X χ y ξ y x y n dy B n,χ,ξ x . 2.3 From 2.3, we derive that B n,χ,ξ x n l0 n l B l,χ,ξ x n−l . 2.4 4 Journal of Inequalities and Applications By 1.5 and 2.3,weseethat X χ x ξ x e xt dx 1 d d−1 a0 χ a e at ξ a Z p ξ dx e dxt dx 1 d d−1 a0 χ a ξ a dt ξ d e dt − 1 e a/ddt . 2.5 From 2.2 and 2.5,weobtainthat X χ x ξ x e xt dx ∞ n0 d n−1 d−1 a0 χ a ξ a B n,ξ d a d t n n! . 2.6 Thus we have the following theorem from 2.1 and 2.6. Theorem 2.1. For ξ ∈ T p , one has B n,χ,ξ d n−1 d−1 a0 χ a ξ a B n,ξ d a d . 2.7 By 1.3 and 1.5, we have that for n ∈ N, X f x n dx X f x dx n−1 i0 f i , 2.8 where f idf x/dx| xi . Taking fxχxξ x e xt in 2.8, it follows that 1 t X χ x ξ ndx e ndx t dx − X χ x ξ x e xt dx nd X χ x ξ x e xt dx X ξ ndx e ndxt dx ξ nd e ndt − 1 ξ d e dt − 1 d−1 i0 χ i ξ i e it . 2.9 Thus, we have 1 t X χ x ξ ndx e ndx t dx − X χ x ξ x e xt dx ∞ k0 nd−1 l0 χ l ξ l l k t k k! . 2.10 For k ∈ Z , let us define the p-adic functional Kχ, ξ, k : n as follows: K χ, ξ, k : n n l0 χ l ξ l l k . 2.11 By 2.10 and 2.11,weseethatfork, n, d ∈ N, X χ x ξ ndx nd x k dx − X χ x ξ x x k dx kK χ, ξ, k − 1:nd − 1 . 2.12 Journal of Inequalities and Applications 5 From 2.3 and 2.12, we have the following result. Theorem 2.2. For ξ ∈ T p and k, n, d ∈ N, one has ξ nd B k,χ,ξ nd − B k,χ,ξ kK χ, ξ, k − 1:nd − 1 . 2.13 Let w 1 ,w 2 ,d ∈ N. Then we have that d X X χ x 1 χ x 2 ξ w 1 x 1 w 2 x 2 e w 1 x 1 w 2 x 2 t dx 1 dx 2 X ξ dw 1 w 2 x e dw 1 w 2 xt dx t ξ dw 1 w 2 e dw 1 w 2 t − 1 ξ w 1 d e dw 1 t − 1 ξ w 2 d e dw 2 t − 1 d−1 a0 χ a ξ w 1 a e w 1 at d−1 b0 χ b ξ w 2 b e w 2 bt . 2.14 By 2.9, 2.10,and2.11,weseethat w 1 d X χ x ξ x e xt dx X ξ dw 1 x e dw 1 xt dx ∞ k0 K χ, ξ, k : dw 1 − 1 t k k! . 2.15 Now let us define the p-adic functional Y χ,ξ w 1 ,w 2 as follows: Y χ,ξ w 1 ,w 2 d X X χ x 1 χ x 2 ξ w 1 x 1 w 2 x 2 e w 1 x 1 w 2 x 2 w 1 w 2 xt dx 1 dx 2 X ξ dw 1 w 2 x 3 e dw 1 w 2 x 3 t dx 3 . 2.16 Then it follows from 2.14 that Y χ,ξ w 1 ,w 2 t ξ dw 1 w 2 e dw 1 w 2 t − 1 e w 1 w 2 xt ξ w 1 d e dw 1 t − 1 ξ w 2 d e dw 2 t − 1 d−1 a0 χ a ξ w 1 a e w 1 at d−1 b0 χ b ξ w 2 b e w 2 bt . 2.17 By 2.15 and 2.16,weobtainthat Y χ,ξ w 1 ,w 2 1 w 1 X χ x 1 ξ w 1 x 1 e w 1 x 1 w 2 x t dx 1 dw 1 X χ x 2 ξ w 2 x 2 e w 2 x 2 t dx 2 X ξ dw 1 w 2 x e dw 1 w 2 xt dx ∞ l0 l i0 l i B i,χ,ξ w 1 w 2 x K χ, ξ w 2 ,l− i : dw 1 − 1 w i−1 1 w l−i 2 t l l! . 2.18 6 Journal of Inequalities and Applications Onthe other hand, thesymmetric property of Y χ,ξ w 1 ,w 2 shows that Y χ,ξ w 1 ,w 2 1 w 2 X χ x 2 ξ w 2 x 2 e w 2 x 2 w 1 x t dx 2 dw 2 X χ x 1 ξ w 1 x 1 e w 1 x 1 t dx 1 X ξ dw 1 w 2 x e dw 1 w 2 xt dx ∞ l0 l i0 l i B i,χ,ξ w 2 w 1 x K χ, ξ w 1 ,l− i : dw 2 − 1 w i−1 2 w l−i 1 t l l! . 2.19 Comparing the coefficients onthe both sides of 2.18 and 2.19,wehavethefollowing theorem. Theorem 2.3. Let ξ ∈ T p and d, w 1 ,w 2 ∈ N. Then one has l i0 l i B i,χ,ξ w 1 w 2 x K χ, ξ w 2 ,l− i : dw 1 − 1 w i−1 1 w l−i 2 l i0 l i B i,χ,ξ w 2 w 1 x K χ, ξ w 1 ,l− i : dw 2 − 1 w i−1 2 w l−i 1 . 2.20 We also derive some identities forthegeneralizedtwistedBernoulli numbers. Taking x 0inTheorem 2.3, we have the following corollary. Corollary 2.4. Let ξ ∈ T p and d, w 1 ,w 2 ∈ N. Then one has l i0 l i B i,χ,ξ w 1 K χ, ξ w 2 ,l−i : dw 1 −1 w i−1 1 w l−i 2 l i0 l i B i,χ,ξ w 2 K χ, ξ w 1 ,l− i : dw 2 − 1 w i−1 2 w l−i 1 . 2.21 Now we will derive another identities forthegeneralizedtwistedBernoulli polynomials using thesymmetric property of Y χ,ξ w 1 ,w 2 .From1.2, 2.15 and 2.17,we see that Y χ,ξ w 1 ,w 2 e w 1 w 2 xt w 1 X χ x 1 ξ w 1 x 1 e w 1 x 1 t dx 1 dw 1 X χ x 2 ξ w 2 x 2 e w 2 x 2 t dx 2 X ξ dw 1 w 2 x e dw 1 w 2 xt dx 1 w 1 dw 1 −1 i0 χ i ξ w 2 i X χ x 1 ξ w 1 x 1 e w 1 x 1 w 2 xw 2 /w 1 it dx 1 ∞ k0 dw 1 −1 i0 χ i ξ w 2 i B k,χ,ξ w 1 w 2 x w 2 w 1 i w k−1 1 t k k! . 2.22 Journal of Inequalities and Applications 7 From thesymmetric property of Y χ,ξ w 1 ,w 2 ,wealsoseethat Y χ,ξ w 1 ,w 2 e w 1 w 2 xt w 2 X χ x 2 ξ w 2 x 2 e w 2 x 2 t dx 2 dw 2 X χ x 1 ξ w 1 x 1 e w 1 x 1 t dx 1 X ξ dw 1 w 2 x e dw 1 w 2 xt dx 1 w 2 dw 2 −1 i0 χ i ξ w 1 i X χ x 2 ξ w 2 x 2 e w 2 x 2 w 1 xw 1 /w 2 it dx 2 ∞ k0 dw 2 −1 i0 χ i ξ w 1 i B k,χ,ξ w 2 w 1 x w 1 w 2 i w k−1 2 t k k! . 2.23 Comparing the coefficients onthe both sides of 2.22 and 2.23, we obtain the following theorem. Theorem 2.5. Let ξ ∈ T p and d, w 1 ,w 2 ∈ N. Then one has dw 1 −1 i0 χ i ξ w 2 i B k,χ,ξ w 1 w 2 x w 2 w 1 i w k−1 1 dw 2 −1 i0 χ i ξ w 1 i B k,χ,ξ w 2 w 1 x w 1 w 2 i w k−1 2 . 2.24 If we take x 0inTheorem 2.5, we also derive the interesting identity forthegeneralizedtwistedBernoulli numbers as follows: for d, w 1 ,w 2 ∈ N, dw 1 −1 i0 χ i ξ w 2 i B k,χ,ξ w 1 w 2 w 1 i w k−1 1 dw 2 −1 i0 χ i ξ w 1 i B k,χ,ξ w 2 w 1 w 2 i w k−1 2 . 2.25 Acknowledgment The present research has been conducted by theresearch grant of the Kwangwoon University in 2009. References 1 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. 2 T. Kim, “On the symmetries of the q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 914367, 7 pages, 2008. 3 T. Kim, “Symmetry p-adic invariant integral on Z p forBernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008. 4 T. Kim, “On the multiple q-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 481–486, 2008. 5 T. Kim, “Note on q-Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 1, pp. 9–15, 2008. 6 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. 7 T. Kim, S H. Rim, and B. Lee, “Some identities of symmetry forthegeneralizedBernoulli numbers and polynomials,” Abstract and Applied Analysis, vol. 2009, Article ID 848943, 8 pages, 2009. 8 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000, 1948. 8 Journal of Inequalities and Applications 9 M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiple p-adic q-L-function of two variables,” Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007. 10 A. S. Hegazi and M. Mansour, “A note on q-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 13, no. 1, pp. 9–18, 2006. 11 Y H.Kim,“Onthep-adic interpolation functions of thegeneralizedtwisted h, q-Euler numbers,” International Journal of Mathematical Analysis, vol. 3, no. 18, pp. 897–904, 2008. 12 Y H.Kim,W.Kim,andL C.Jang,“Ontheq-extension of Apostol-Euler numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 296159, 10 pages, 2008. 13 Y H. Kim and K W. Hwang, “Symmetry of power sum and twistedBernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 127–133, 2009. 14 B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, pp. 412–422, 2005. 15 Y. Simsek, “Theorems ontwisted L-function and twistedBernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005. 16 Y. Simsek, “On p-adic twisted q-L-functions related to generalizedtwistedBernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006. 17 Y. Simsek, “Generating functions of thetwistedBernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251– 278, 2008. 18 Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of thetwisted h, q-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44–56, 2007. 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 K. T. Atanassov and M. V. Vassilev-Missana, “On one of Murthy-Ashbacher’s conjectures related to Euler’s totient function,” Proceedings of the Jangjeon Mathematical Society, vol. 9, no. 1, pp. 47–49, 2006. 21 T. Kim, “A note on some formulae forthe q-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 9, no. 2, pp. 227–232, 2006. 22 T. Kim, “Note on Dedekind type DC sums,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 249–260, 2009. 23 T. Kim, “A note onthegeneralized q-Euler numbers,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 1, pp. 45–50, 2009. 24 T. Kim, “New approach to q-Euler, Genocchi numbers and their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009. 25 T. Kim, “Symmetry identities forthetwistedgeneralized Euler polynomials,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 2, pp. 151–155, 2009. 26 Y H. Kim, W. Kim, and C. S. Ryoo, “On thetwisted q-Euler zeta function associated with twisted q-Euler numbers,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 1, pp. 93–100, 2009. 27 H. Ozden, I. N. Cangul, and Y. S imsek, “Remarks on q-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009. . Corporation Journal of Inequalities and Applications Volume 2009, Article ID 164743, 8 pages doi:10.1155/2009/164743 Research Article On the Symmetric Properties for the Generalized Twisted Bernoulli. the symmetry for the generalized twisted Bernoulli polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Bernoulli polynomials using the symmetric. generalized twisted Bernoulli polynomials and numbers attached to χ.InSection 2, we give interesting identities for the power sums and the generalized twisted Bernoulli polynomials using the symmetric properties