Báo cáo hóa học: " Research Article On the Identities of Symmetry for the ζ-Euler Polynomials of Higher Order" pptx

Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 273545, 9 pages doi:10.1155/2009/273545 Research Article On the Identities of Symmetry for the ζ-Euler Polynomials of 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 of symmetry for the multivariate p-adic fermionic integral on Z p . From these symmetries, we can derive 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 of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher 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 .LetUDZ p  be the space of uniformly differentiable functions on Z p . For f ∈ UDZ 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  x0 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.From1.2, we have I −1  f 1   I −1  f   2f  0  1.3 see 9, 10, where f 1 xfx  1. For ζ ∈ C p with |1 − ζ| p < 1, let fxe xt ζ x . Then, we define the ζ-Euler numbers as follows:  Z p ζ x e xt dμ −1  x   2 ζe t  1  ∞  n0 E n,ζ t n n! , 1.4 where E n,ζ are called the ζ-Euler numbers. We can show that 2 ζe t  1  1  ζ −1 e t  ζ −1 · 2 1  ζ  2 1  ζ ∞  n0 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 the ζ-Euler polynomials as follows: 2 ζe t  1 e xt  ∞  n0 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 ∞  n0 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 ∞  k0 T k,ζ  n  t k k!  ∞  k0  n  0  −1    k ζ   t k k!  n  0  −1   ζ  e t  1   −1  n1 e  n1  t ζe t  1 . 1.10 Advances in Difference Equations 3 The ζ-Euler polynomials of 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  ∞  n0 E k n,ζ  x  t n n! . 1.11 Then the values of E k n,ζ x at x  0 are called the ζ-Euler numbers of order k. When k  1, the polynomials or numbers are called the ζ-Euler polynomials or numbers. The purpose of this paper is to investigate some 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 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 of higher order. 2. On the Symmetry for the ζ-Euler Polynomials of Higher Order Let w 1 ,w 2 ∈ N with w 1 ≡ 1mod 2 and w 2 ≡ 1mod 2. Then we set R m  w 1 ,w 2    Z m p e w 1 x 1 x 2 ···x m w 2 xt 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 yt 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 .From2.1 ,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  ∞  k0  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  ∞  n0 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,and2.5, we can derive R m  w 1 ,w 2    ∞  0 E m ,ζ w 1  w 2 x  w  1 t  !  ∞  k0 T k,ζ w 2  w 1 − 1  w k 2 k! t k  ∞  i0 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  !  ⎛ ⎝ ∞  j0 ⎛ ⎝ j  k0 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  !  ⎛ ⎝ ∞  j0  j  k0 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!  ∞  n0 ⎛ ⎝ n  j0  j  k0 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!  ∞  n0 ⎛ ⎝ n  j0  n j  w j 2 w n−j 1 E m n−j,ζ w 1  w 2 x  j  k0 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  !  ∞  k0 T k,ζ w 1  w 2 − 1  w k 1 t k k!  ∞  i0 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  !  ⎛ ⎝ ∞  j0 ⎛ ⎝ j  k0 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  !  ⎛ ⎝ ∞  j0 ⎛ ⎝ j  k0 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  !  ⎛ ⎝ ∞  j0  j  k0  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! ⎞ ⎠  ∞  n0 ⎛ ⎝ n  j0  j  k0  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!  ∞  n0 ⎛ ⎝ n  j0  n j  w j 1 w n−j 2 E m n−j,ζ w 2  w 1 x  j  k0  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 ≡ 1mod 2 , w 2 ≡ 1mod 2 , and n ≥ 0,m ≥ 1, one has n  j0  n j  w j 2 w n−j 1 E m n−j,ζ w 1  w 2 x  j  k0 T k,ζ w 2  w 1 − 1   j k  E m−1 j−k,ζ w 2  w 1 y   n  j0  n j  w j 1 w n−j 2 E m n−j,ζ w 2  w 1 x  j  k0  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  1in2.9. Then we have n  j0  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  j0  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  i0  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  i0  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  1in2.11, then we have E n,ζ  w 1 x   n  i0  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  i0  −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  i0  −1  i ζ w 2 i  Z m p e w 1 x 1 ···x m w 2 /w 1 iw 2 xt 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 yt ζ w 2 x 1 ···w 2 x m−1 dμ −1  x 1  ···dμ −1  x m−1     w 1 −1  i0  −1  i ζ w 2 i ∞  k0 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  !   ∞  n0  n  k0  w 1 −1  i0  −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!  ∞  n0  n  k0  n k  w k 1 w n−k 2 E  m−1  n−k,ζ w 2  w 1 y  w 1 −1  i0  −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  i0  −1  i e w 1 it ζ w 1 i  ×   Z m−1 p e w 1 x 1 ···x m−1 w 2 yt ζ w 1 x 1 ···x m−1  dμ −1  x 1  ···dμ −1  x m−1    w 2 −1  i0  −1  i ζ w 1 i   Z m p e w 2 x 1 ···x m w 1 /w 2 iw 1 xt 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 yt ζ w 1 x 1 ···w 1 x m−1 dμ −1  x 1  ···dμ −1  x m−1     w 2 −1  i0  −1  i ζ w 1 i ∞  k0 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  ∞  n0  n  k0  w 2 −1  i0  −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!  ∞  n0  n  k0  n k  w k 2 w n−k 1 E m−1 n−k,ζ w 1  w 2 y  w 2 −1  i0  −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 ≡ 1mod 2  and w 2 ≡ 1mod 2 , one has n  k0  n k  w k 1 w n−k 2 E m−1 n−k,ζ w 2  w 1 y  w 1 −1  i0  −1  i ζ w 2 i E m k,ζ w 1  w 2 x  w 2 w 1 i   n  k0  n k  w k 2 w n−k 1 E m−1 n−k,ζ w 1  w 2 y  w 2 −1  i0  −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  i0  −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  i0  −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  i0  −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 the research grant of the Kwangwoon University in 2009. References 1 T. Kim, “Symmetry p-adic invariant integral on Z p for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008. 2 T. 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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