Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 273545, 9 pages doi:10.1155/2009/273545 ResearchArticleOntheIdentitiesofSymmetryfortheζ-EulerPolynomialsof H igher Order Taekyun Kim, 1 Kyoung Ho Park, 2 and Kyung-won Hwang 3 1 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea 2 Department of Mathematics, Sogang University, Seoul 121-742, South Korea 3 Department of General Education, Kookmin University, Seoul 139-702, South Korea Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr Received 19 February 2009; Revised 31 May 2009; Accepted 18 June 2009 Recommended by Agacik Zafer The main purpose of this paper is to investigate several further interesting properties ofsymmetryforthe multivariate p-adic fermionic integral on Z p . From these symmetries, we can derive some recurrence identitiesfortheζ-Eulerpolynomialsofhigher order, which are closely related to the Frobenius-Euler polynomialsofhigher order. By using our identitiesofsymmetryforthe ζ- Euler polynomialsofhigher order, we can obtain many identities related to the Frobenius-Euler polynomialsofhigher order. 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/Definition Let p be a fixed odd prime number. Throughout this paper, Z p , Q p , C, and C p will, respectively, denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Q p .Letv p be the normalized exponential valuation of C p with |p| p p −v p p p −1 .LetUDZ p be the space of uniformly differentiable functions on Z p . For f ∈ UDZ p , q ∈ C p with |1 − q| p < 1, the fermionic p-adic q-integral on Z p is defined as I −q f Z p f x dμ −q x lim N →∞ 1 q 1 q p N p N −1 x0 f x −q x 1.1 see 1. Let us define the fermionic p-adic invariant integral on Z p as follows: I −1 f lim q → 1 I −q f Z p f x dμ −1 x 1.2 2 Advances in Difference Equations see 1–8.From1.2, we have I −1 f 1 I −1 f 2f 0 1.3 see 9, 10, where f 1 xfx 1. For ζ ∈ C p with |1 − ζ| p < 1, let fxe xt ζ x . Then, we define theζ-Euler numbers as follows: Z p ζ x e xt dμ −1 x 2 ζe t 1 ∞ n0 E n,ζ t n n! , 1.4 where E n,ζ are called theζ-Euler numbers. We can show that 2 ζe t 1 1 ζ −1 e t ζ −1 · 2 1 ζ 2 1 ζ ∞ n0 H n −ζ −1 t n n! , 1.5 where H n −ζ −1 are the Frobenius-Euler numbers. By comparing the coefficients on both sides of 1.4 and 1.5,weseethat E n,ζ 2 1 ζ H n −ζ −1 . 1.6 Now, we also define theζ-Eulerpolynomials as follows: 2 ζe t 1 e xt ∞ n0 E n,ζ x t n n! . 1.7 In the viewpoint of 1.5, we can show that 2 ζe t 1 e xt e xt 1 ζ −1 e t ζ −1 · 2 1 ζ 2 1 ζ ∞ n0 H n −ζ −1 ,x t n n! , 1.8 where H n −ζ −1 ,x are the nth Frobenius-Euler polynomials. From 1.7 and 1.8,wenote that E n,ζ x 2 1 ζ H n −ζ −1 ,x 1.9 cf. 1–8, 11–18. For each positive integer k,letT k,ζ n n 0 −1 ζ k . Then we have ∞ k0 T k,ζ n t k k! ∞ k0 n 0 −1 k ζ t k k! n 0 −1 ζ e t 1 −1 n1 e n1 t ζe t 1 . 1.10 Advances in Difference Equations 3 Theζ-Eulerpolynomialsof order k, denoted E k n,ζ x, are defined as e xt 2 ζe t 1 k 2 ζe t 1 ×···× 2 ζe t 1 e xt ∞ n0 E k n,ζ x t n n! . 1.11 Then the values of E k n,ζ x at x 0 are called theζ-Euler numbers of order k. When k 1, thepolynomials or numbers are called theζ-Eulerpolynomials or numbers. The purpose of this paper is to investigate some properties ofsymmetryforthe multivariate p-adic fermionic integral on Z p . From the properties ofsymmetryforthe multivariate p-adic fermionic integral on Z p , we derive some identitiesofsymmetryfortheζ-Eulerpolynomialsofhigher order. By using our identitiesofsymmetryfortheζ-Eulerpolynomialsofhigher order, we can obtain many identities related to the Frobenius-Euler polynomialsofhigher order. 2. OntheSymmetryfortheζ-EulerPolynomialsofHigher Order Let w 1 ,w 2 ∈ N with w 1 ≡ 1mod 2 and w 2 ≡ 1mod 2. Then we set R m w 1 ,w 2 Z m p e w 1 x 1 x 2 ···x m w 2 xt ζ w 1 x 1 ···w 1 x m dμ −1 x 1 ···dμ −1 x m Z p ζ w 1 w 2 x e w 1 w 2 xt dμ −1 x × Z m p e w 2 x 1 x 2 ···x m w 1 yt ζ w 2 x 1 ···w 2 x m dμ −1 x 1 ···dμ −1 x m , 2.1 where Z m p f x 1 , ,x m dμ −1 x 1 ···dμ −1 x m Z p ··· Z p f x 1 , ,x m dμ −1 x 1 ···dμ −1 x m . 2.2 Thus, we note that this expression for R m w 1 ,w 2 is symmetry in w 1 and w 2 .From2.1 ,we have R m w 1 ,w 2 Z m p e w 1 x 1 ···x m t ζ w 1 x 1 ···w 1 x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt × ⎛ ⎝ Z p e w 2 x m t ζ w 2 x m dμ −1 x m Z p e w 1 w 2 xt ζ w 1 w 2 x dμ −1 x ⎞ ⎠ × Z m−1 p e w 2 x 1 ···x m−1 t ζ w 2 x 1 ···w 2 x m−1 dμ −1 x 1 ···dμ −1 x m−1 e w 1 w 2 yt . 2.3 4 Advances in Difference Equations We can show that Z p e xt ζ x dμ −1 x Z p e w 1 xt ζ w 1 x dμ −1 x w 1 −1 0 −1 ζ e t ∞ k0 T k,ζ w 1 − 1 t k k! . 2.4 By 1.4 and 1.11,weseethat Z m p e w 1 x 1 ···x m t ζ w 1 x 1 ···w 1 x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt 2 ζ w 1 e w 1 t 1 m e w 1 w 2 xt ∞ n0 E m n,ζ w 1 w 2 x w n 1 t n n! . 2.5 Thus, we have E m n,ζ w 1 w 2 x n 0 n E m ,ζ w 1 w n− 2 x n− . 2.6 From 2.3, 2.4,and2.5, we can derive R m w 1 ,w 2 ∞ 0 E m ,ζ w 1 w 2 x w 1 t ! ∞ k0 T k,ζ w 2 w 1 − 1 w k 2 k! t k ∞ i0 E m−1 i,ζ w 2 w 1 y w i 2 i! t i ∞ 0 E m ,ζ w 1 w 2 x w 1 t ! ⎛ ⎝ ∞ j0 ⎛ ⎝ j k0 T k,ζ w 2 w 1 − 1 w k 2 w j−k 2 E m−1 j−k w 1 y k! j − k ! j! ⎞ ⎠ t j j! ⎞ ⎠ ∞ 0 E m ,ζ w 1 w 2 x w 1 t ! ⎛ ⎝ ∞ j0 j k0 T k,ζ w 2 w 1 − 1 j k E m−1 j−k,ζ w 2 w 1 y w j 2 ⎞ ⎠ t j j! ∞ n0 ⎛ ⎝ n j0 j k0 T k,ζ w 2 w 1 − 1 j k E m−1 j−k,ζ w 2 w 1 y w j 2 w n−j 1 n − j !j! E m n−j,ζ w 1 w 2 x n! ⎞ ⎠ t n n! ∞ n0 ⎛ ⎝ n j0 n j w j 2 w n−j 1 E m n−j,ζ w 1 w 2 x j k0 T k,ζ w 2 w 1 − 1 j k E m−1 j−k,ζ w 2 w 1 y ⎞ ⎠ t n n! . 2.7 Advances in Difference Equations 5 By the same method, we also see that R m w 1 ,w 2 Z m p e w 2 x 1 ···x m t ζ w 2 x 1 ···w 2 x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt × ⎛ ⎝ Z p e w 1 x m t ζ w 1 x m dμ −1 x m Z p e w 1 w 2 xt ζ w 1 w 2 x dμ −1 x ⎞ ⎠ × Z m−1 p e w 1 x 1 ···x m−1 t ζ w 1 x 1 ···w 1 x m−1 dμ −1 x 1 ···dμ −1 x m−1 e w 1 w 2 yt ∞ 0 E m ,ζ w 2 w 1 x w 2 t ! ∞ k0 T k,ζ w 1 w 2 − 1 w k 1 t k k! ∞ i0 E m−1 i,ζ w 1 w 2 y w i 1 t i i! ∞ 0 E m ,ζ w 2 w 1 x w 2 t ! ⎛ ⎝ ∞ j0 ⎛ ⎝ j k0 T k,ζ w 1 w 2 − 1 k! E m−1 j−k w 2 y j − k ! ⎞ ⎠ w j 1 t j ⎞ ⎠ ∞ 0 E m ,ζ w 2 w 1 x w 2 t ! ⎛ ⎝ ∞ j0 ⎛ ⎝ j k0 T k,ζ w 1 w 2 − 1 E m−1 j−k w 2 y k! j − k ! j! ⎞ ⎠ w j 1 t j j! ⎞ ⎠ ∞ 0 E m ,ζ w 2 w 1 x w 2 t ! ⎛ ⎝ ∞ j0 j k0 j k T k,ζ w 1 w 2 − 1 E m−1 j−k,ζ w 1 w 2 y w j 1 t j j! ⎞ ⎠ ∞ n0 ⎛ ⎝ n j0 j k0 j k T k,ζ w 1 w 2 − 1 E m−1 j−k,ζ w 1 w 2 y w j 1 w n−j 2 j! n − j ! E m n−j,ζ w 2 w 1 x n! ⎞ ⎠ t n n! ∞ n0 ⎛ ⎝ n j0 n j w j 1 w n−j 2 E m n−j,ζ w 2 w 1 x j k0 j k T k,ζ w 1 w 2 − 1 E m−1 j−k,ζ w 1 w 2 y ⎞ ⎠ t n n! . 2.8 By comparing the coefficients on both sides of 2.7 and 2.8, we obtain the following. Theorem 2.1. For w 1 ,w 2 ∈ N with w 1 ≡ 1mod 2 , w 2 ≡ 1mod 2 , and n ≥ 0,m ≥ 1, one has n j0 n j w j 2 w n−j 1 E m n−j,ζ w 1 w 2 x j k0 T k,ζ w 2 w 1 − 1 j k E m−1 j−k,ζ w 2 w 1 y n j0 n j w j 1 w n−j 2 E m n−j,ζ w 2 w 1 x j k0 j k T k,ζ w 1 w 2 − 1 E m−1 j−k,ζ w 1 w 2 y . 2.9 6 Advances in Difference Equations Let y 0andm 1in2.9. Then we have n j0 n j w n−j 1 w j 2 E n−j,ζ w 1 w 2 x T k,ζ w 2 w 1 − 1 n j0 n j w j 1 w n−j 2 E n−j,ζ w 2 w 1 x T k,ζ w 1 w 2 − 1 . 2.10 From 2.10,wenotethat n i0 n i w i 1 w n−i 2 E i,ζ w 1 w 2 x T n−i,ζ w 2 w 1 − 1 n i0 n i w n−i 1 w i 2 E i,ζ w 2 w 1 x T n−i,ζ w 1 w 2 − 1 . 2.11 If we take w 2 1in2.11, then we have E n,ζ w 1 x n i0 n i w i 1 E i,ζ w 1 x T n−i,ζ w 1 − 1 . 2.12 From 2.3,wenotethat R m w 1 ,w 2 Z m p e w 1 x 1 ···x m t ζ w 1 x 1 ···w 1 x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt × ⎛ ⎝ Z p e w 2 x m t ζ w 2 x m dμ −1 x m Z p e w 1 w 2 xt ζ w 1 w 2 x dμ −1 x ⎞ ⎠ × Z m−1 p e w 2 x 1 ···x m−1 t ζ w 2 x 1 ···w 2 x m−1 dμ −1 x 1 ···dμ −1 x m−1 e w 1 w 2 yt Z m p e w 1 x 1 ···x m t ζ w 1 x 1 ···w 1 x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt × w 1 −1 i0 −1 i e w 2 it ζ w 2 i × Z m−1 p e w 2 x 1 ···x m−1 t ζ w 2 x 1 ···w 2 x m−1 dμ −1 x 1 ···dμ −1 x m−1 e w 1 w 2 yt Advances in Difference Equations 7 w 1 −1 i0 −1 i ζ w 2 i Z m p e w 1 x 1 ···x m w 2 /w 1 iw 2 xt ζ w 1 x 1 ···w 1 x m dμ −1 x 1 ···dμ −1 x m × Z m−1 p e w 2 x 1 ···x m−1 w 1 yt ζ w 2 x 1 ···w 2 x m−1 dμ −1 x 1 ···dμ −1 x m−1 w 1 −1 i0 −1 i ζ w 2 i ∞ k0 E m k,ζ w 1 w 2 w 1 i w 2 x w k 1 t k k! ∞ 0 E m−1 ,ζ w 2 w 1 y w 2 t ! ∞ n0 n k0 w 1 −1 i0 −1 i ζ w 2 i E m k,ζ w 1 w 2 x w 2 w 1 i w k 1 k! E m−1 n−k,ζ w 2 w 1 y w n−k 2 n − k ! n! t n n! ∞ n0 n k0 n k w k 1 w n−k 2 E m−1 n−k,ζ w 2 w 1 y w 1 −1 i0 −1 i ζ w 2 i E m k,ζ w 1 w 2 x w 2 w 1 i t n n! . 2.13 By the symmetric property of R m w 1 ,w 2 in w 1 ,w 2 ,wealsoseethat R m w 1 ,w 2 Z m p e w 2 x 1 ···x m t ζ w 2 x 1 ···x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt × ⎛ ⎝ Z p e w 1 x m t ζ w 1 x m dμ −1 x m Z p e w 1 w 2 xt ζ w 1 w 2 x dμ −1 x ⎞ ⎠ × Z m−1 p e w 1 x 1 ···x m−1 t ζ w 1 x 1 ···x m−1 dμ −1 x 1 ···dμ −1 x m−1 e w 1 w 2 yt Z m p e w 2 x 1 ···x m t ζ w 2 x 1 ···x m dμ −1 x 1 ···dμ −1 x m e w 1 w 2 xt × w 2 −1 i0 −1 i e w 1 it ζ w 1 i × Z m−1 p e w 1 x 1 ···x m−1 w 2 yt ζ w 1 x 1 ···x m−1 dμ −1 x 1 ···dμ −1 x m−1 w 2 −1 i0 −1 i ζ w 1 i Z m p e w 2 x 1 ···x m w 1 /w 2 iw 1 xt ζ w 2 x 1 ···x m dμ −1 x 1 ···dμ −1 x m × Z m−1 p e w 1 x 1 ···x m−1 w 2 yt ζ w 1 x 1 ···w 1 x m−1 dμ −1 x 1 ···dμ −1 x m−1 w 2 −1 i0 −1 i ζ w 1 i ∞ k0 E m k,ζ w 2 w 1 w 2 i w 1 x w k 2 t k k! ∞ 0 E m−1 ,ζ w 1 w 2 y w 1 t ! 8 Advances in Difference Equations ∞ n0 n k0 w 2 −1 i0 −1 i ζ w 1 i E m k,ζ w 2 w 1 x w 1 w 2 i w k 2 k! E m−1 n−k w 2 y w n−k 1 n − k ! n! t n n! ∞ n0 n k0 n k w k 2 w n−k 1 E m−1 n−k,ζ w 1 w 2 y w 2 −1 i0 −1 i ζ w 1 i E m k,ζ w 2 w 1 x w 1 w 2 i t n n! . 2.14 By comparing the coefficients on both sides of 2.13 and 2.14, we obtain the following theorem. Theorem 2.2. For w 1 ,w 2 ∈ N with w 1 ≡ 1mod 2 and w 2 ≡ 1mod 2 , one has n k0 n k w k 1 w n−k 2 E m−1 n−k,ζ w 2 w 1 y w 1 −1 i0 −1 i ζ w 2 i E m k,ζ w 1 w 2 x w 2 w 1 i n k0 n k w k 2 w n−k 1 E m−1 n−k,ζ w 1 w 2 y w 2 −1 i0 −1 i ζ w 1 i E m k,ζ w 2 w 1 x w 1 w 2 i . 2.15 Let y 0andm 1, we have w n 1 w 1 −1 i0 −1 i ζ w 2 i E n,ζ w 1 w 2 x w 2 w 1 i w n 2 w 2 −1 i0 −1 i ζ w 1 i E n,ζ w 2 w 1 x w 1 w 2 i . 2.16 From 2.16, we can derive w 1 −1 i0 −1 i ζ i E n,ζ w 1 x 1 w 1 i 1 w n 1 E n,ζ w 1 x . 2.17 Acknowledgment The present research has been conducted by theresearch grant ofthe Kwangwoon University in 2009. References 1 T. 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Tuenter, “A symmetryof power sum polynomials and Bernoulli numbers,” The American Mathematical Monthly, vol. 108, no. 3, pp. 258–261, 2001. . symmetry for the ζ-Euler polynomials of higher order. By using our identities of symmetry for the ζ-Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials. some recurrence identities for the ζ-Euler polynomials of higher order, which are closely related to the Frobenius-Euler polynomials of higher order. By using our identities of symmetry for the ζ- Euler polynomials. properties of symmetry for the multivariate p-adic fermionic integral on Z p . From the properties of symmetry for the multivariate p-adic fermionic integral on Z p , we derive some identities of symmetry