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Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 738603, 11 pages doi:10.1155/2008/738603 Research ArticleMultipleTwistedq-EulerNumbersandPolynomialsAssociatedwithp-Adic q-Integrals Lee-Chae Jang Department of Mathematics and Computer Science, Konkuk University, Chungju 380701, South Korea Correspondence should be addressed to Lee-Chae Jang, leechae.jang@kku.ac.kr Received 14 January 2008; Revised 25 February 2008; Accepted 26 February 2008 Recommended by Martin Bohner By using p-adic q-integrals on Z p ,wedefinemultipletwistedq-Euler numbersand polynomials. We also find Witt’s type formula for multipletwistedq-Eulernumbersand discuss some characterizations of multipletwistedq-Euler Zeta functions. In particular, we construct multipletwisted Barnes’ type q-Eulerpolynomialsandmultipletwisted Barnes’ type q-Euler Zeta functions. Finally, we define multipletwisted Dirichlet’s type q-Eulernumbersand polynomials, and give Witt’s type formula for them. Copyright q 2008 Lee-Chae Jang. 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 odd prime number. Throughout this paper, Z p , Q p ,andC p are, respectively, the ring of p-adic rational integers, the field of p-adic rational numbers, and the p-adic completion of the algebraic closure of Q p .Thep-adic absolute value in C p is normalized so that |p| p 1/p. When one talks about q-extension, q is variously considered as an indeterminate, a complex number, q ∈ C or a p-adic number q ∈ C p .Ifq ∈ C, one normally assumes that |q| < 1. If q ∈ C p , one normally assumes that |1 − q| p <p −1/p−1 so that q x expx log q for each x ∈ Z p . We use the notations x q 1 − q x 1 − q , x −q 1 − −q x 1 q 1.1 cf. 1–14, for all x ∈ Z p . For a fixed odd positive integer d with p, d1, set X X d lim ← n Z/dp n Z,X 1 Z p , 2 Advances in Difference Equations X ∗ 0<a<dp a,p1 a dpZ p , a dp n Z p x ∈ X | x ≡ a mod dp n , 1.2 where a ∈ Z lies in 0 ≤ a<dp n . For any n ∈ N, μ q a dp n Z p q a dp n q 1.3 is known to be a distribution on X cf. 1–28. We say that f is uniformly differentiable function at a point a ∈ Z p and denote this property by f ∈ UD Z p if the difference quotients F f x, y fx − fy x − y 1.4 have a limit l f a as x, y → a, acf. 25. The p-adic q-integral of a function f ∈ UD Z p was defined as I q f Z p fxdμ q x lim n→∞ 1 p n q p n −1 x0 fxq x , 1.5 I −q f Z p fxdμ −q x lim n→∞ 1 p n q p n −1 x0 fx−q x , 1.6 cf. 4, 24, 25, 28,from1.6,wederive qI −q f 1 I −q f2 q f0, 1.7 where f 1 xfx 1.Ifwetakefxe tx ,thenwehavef 1 xe tx1 e tx e t .From1.7, we obtain that I −q e tx 2 q qe t 1 . 1.8 In Section 2, we define the multipletwistedq-Eulernumbersandpolynomials on Z p and find Witt’s type formula for multipletwistedq-Euler numbers. We also have sums of consecutive multipletwistedq-Euler numbers. In Section 3, we consider multipletwisted q- Euler Zeta functions which interpolate new multipletwistedq-Eulerpolynomials at negative integers and investigate some characterizations of them. In Section 4, we construct the multipletwisted Barnes’ type q-Eulerpolynomialsandmultipletwisted Barnes’ type q-Euler Zeta functions which interpolate new multipletwisted Barnes’ type q-Eulerpolynomials at negative integers. In Section 5, we define multipletwisted Dirichlet’s type q-Eulernumbersandpolynomialsand give Witt’s type formula for them. Lee-Chae Jang 3 2. Multipletwistedq-Eulernumbersandpolynomials In this section, we assume that q ∈ C p with |1 − q| p < 1. For n ∈ N, by the definition of p-adic q-integral on Z p ,wehave q n I −q f n −1 n−1 I −q f2 q n−1 x0 −1 n−1−x q x fx, 2.1 where f n xfx n.Ifn is odd positive integer, we have q n I −q f n I −q f2 q n−1 x0 −1 n−1−x q x fx. 2.2 Let T p ∪ n≥1 C p n lim n→∞ C p n C p ∞ be the locally constant space, where C p n {w | w p n 1} is the cyclic group of order p n .Forw ∈ T p , we denote the locally constant function by φ w : Z p −→ C p ,x−→ w x , 2.3 cf. 5, 7–14, 16, 18.Ifwetakefxφ w xe tx ,thenwehave Z p e tx φ w xdμ −q x 2 q qwe t 1 . 2.4 Now we define the twistedq-Eulernumbers E q n,w as follows: F w t 2 q qwe t 1 ∞ no E q n,w t n n! . 2.5 We note that by substituting w 1, lim q→1 E q n,1 E n are the familiar Euler numbers. Over five decades ago, Carlitz defined q-extension of Euler numbers cf. 15.From2.4 and 2.5,we note that Witt’s type formula for a twistedq-Euler number is given by Z p x n w x dμ −q xE q n,w . 2.6 for each w ∈ T p and n ∈ N. Twistedq-Eulerpolynomials E q n,w x are defined by means of the generating function F q w t, x 2 q qwe t 1 e xt ∞ n0 E q n,w x t n n! , 2.7 where E q n,w 0E q n,w . By using the hth iterative fermionic p-adic q-integral o n Z p , we define multipletwistedq-Euler number as follows: Z p ··· Z p h-times w x 1 ···x h e x 1 x 2 ···x h t dμ −q x 1 ···dμ −q x h 2 q qwe t 1 h ∞ n0 E h,q n,w t n n! . 2.8 Thus we give Witt’s type formula for multipletwistedq-Eulernumbers as follows. 4 Advances in Difference Equations Theorem 2.1. For each w ∈ T p and h, n ∈ N, Z p ··· Z p h-times w x 1 ···x h x 1 ··· x h n dμ −q x 1 ···dμ −q x h E h,q n,w , 2.9 where x 1 ··· x h n l 1 ···l h n l 1 , ,l h ≥0 n! l 1 ! ···l h ! x l 1 1 ···x l h h . 2.10 From 2.8 and 2.9, we o btain the following theorem. Theorem 2.2. For w ∈ T p and h, k ∈ N, E h,q k,w l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! E q l 1 ,w ···E q l h ,w . 2.11 From these formulas, we consider multivariate fermionic p-adic q-integral on Z p as follows: Z p ··· Z p h-times w x 1 ···x h e x 1 ···x h xt dμ −q x 1 ···dμ −q x h 2 q qwe t 1 ··· 2 q qwe t 1 e xt 2 q qwe t 1 h e xt . 2.12 Then we can define the multipletwistedq-Eulerpolynomials E h,q n,w x as follows: F h,q w t, x 2 q qwe t 1 h e xt ∞ n0 E h,q n,w x t n n! . 2.13 From 2.12 and 2.13,wenotethat ∞ n0 Z p ··· Z p h-times w x 1 ···x h x 1 ··· x h x n dμ −q x 1 ···dμ −q x h t n n! ∞ n0 E h,q n,w x t n n! . 2.14 Then by the kth differentiation on both sides of 2.14, we obtain the following. Theorem 2.3. For each w ∈ T p and k, h ∈ N, Z p ··· Z p h-times w x 1 ···x h x 1 ··· x h x k dμ −q x 1 ···dμ −q x h E h,q k,w x. 2.15 Lee-Chae Jang 5 Note that x 1 ··· x h x n l 1 ···l h n l 1 , ,l h ≥0 n! l 1 ! ···l h ! x l 1 1 · x l 2 2 ···x h x l h . 2.16 Then we see that Z p ··· Z p h-times w x 1 ···x h x 1 ··· x h x k dμ −q x 1 ···dμ −q x h l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! Z p w x 1 x l 1 1 dμ −q x 1 ··· Z p w x h−1 x l h−1 h−1 dμ −q x h−1 Z p x x h l h dμ −q x h l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! E q l 1 ,w ···E q l h−1 ,w E q l h ,w x. 2.17 From 2.15 and 2.17, we obtain the sums of powers of consecutive q-Eulernumbers as follows. Theorem 2.4. For each w ∈ T p and k, h ∈ N, E h,q k,w x l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! E q l 1 ,w ···E q l h−1 ,w · E q l h ,w x. 2.18 3. Multipletwistedq-Euler Zeta functions For q ∈ C with |q| < 1andw ∈ T p , the multipletwistedq-Eulernumbers can be considered as follows: F h w t 2 q qwe t 1 h ∞ n0 E h,q n,w t n n! , t logqw <π. 3.1 From 3.1,wenotethat ∞ n0 E h,q n,w t n n! F h w t 2 q qwe t 1 h 2 h q 2 q qwe t 1 ··· 2 q qwe t 1 2 h q ∞ n 1 0 −1 n 1 q n 1 w n 1 e n 1 t ··· ∞ n h 0 −1 n h q n h w n h e n h t 2 h q n 1 , ,n h 0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h e n 1 ···n h t . 3.2 6 Advances in Difference Equations By the kth differentiation on both sides of 3.2 at t 0, we obtain that E h,q k,w 2 h q n 1 ···n h / 0 n 1 , ,n h ≥0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h n 1 ··· n h k . 3.3 From 3.3, we derive multipletwistedq-Euler Zeta function as follows: ζ h,q w s2 h q n 1 ···n h / 0 n 1 , ,n h ≥0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h n 1 ··· n h s 3.4 for all s ∈ C. We also obtain the following theorem in which multipletwistedq-Euler Zeta functions interpolate multipletwistedq-Euler polynomials. Theorem 3.1. For w ∈ T p and k, h ∈ N, ζ h,q w −kE h,q k,w . 3.5 4. Multipletwisted Barnes’ type q-Eulerpolynomials In this section, we consider the generating function of multipletwistedq-Euler polynomials: F h w t, x 2 q qwe t 1 h e xt ∞ n0 E h,q n,w x t n n! , t logqw <π, Rex > 0. 4.1 We note that ∞ n0 E h,q n,w x t n n! F h w t, x 2 h q n 1 , ,n h 0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h e n 1 ···n h xt . 4.2 By the kth differentiation on both sides of 4.2 at t 0, we obtain that E h,q k,w x 2 h q n 1 , ,n h 0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h n 1 ··· n h x k . 4.3 Thus we can consider multipletwisted Hurwitz’s type q-Euler Zeta function as follows: ζ h,q w s, x2 h q n 1 ···n h / 0 n 1 , ,n h ≥0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h n 1 ··· n h x s 4.4 for all s ∈ C and Rex > 0. We note that ζ h,q w s, x is analytic function in the whole complex s-plane and ζ h,q w s, 0ζ h,q w s. We also remark that if w 1andh 1, then ζ 1,q 1 s, x ζ q s, x is Hurwitz’s type q-Euler Zeta function see 7, 27. The following theorem means that multipletwistedq-Euler Zeta functions interpolate multipletwistedq-Eulerpolynomials at negative integers. Lee-Chae Jang 7 Theorem 4.1. For w ∈ T p , k, h ∈ N, s ∈ C,andRex > 0, ζ h,q w −k, xE h,q k,w x. 4.5 Let us consider F h w a 1 , ,a h | t, x 2 q qwe a 1 t 1 ··· 2 q qwe a h t 1 e xt 2 h q ∞ n 1 , ,n h 0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h e a 1 n 1 ···a h n h xt ∞ n0 E h,q n,w a 1 , ,a h | x t n n! , 4.6 where a 1 , ,a h ∈ C and max 1≤i≤k {| logq a i t|} <π.ThenE h,q n,w a 1 , ,a h | x will be called multipletwisted Barnes’ type q-Euler polynomials. We note that E h,q n,w 1, 1, ,1 | xE h,q n,w x. 4.7 By the kth differentiation of both sides of 4.6, we obtain the following theorem. Theorem 4.2. For each w ∈ T p , a 1 , ,a h ∈ C, k, h ∈ N,andRex > 0, E h,q k,w a 1 , ,a h | x 2 h q n 1 ···n h / 0 n 1 , ,n h ≥0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h a 1 n 1 ··· a h n h x k , 4.8 where a 1 n 1 ··· a h n h x k l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! a l 1 1 ···a l h−1 h−1 n l 1 1 ···n l h−1 h−1 a h n h x l h . 4.9 From 4.8, we consider multipletwisted Barnes’ type q-Euler Zeta function defined as follows: for each w ∈ T p , a 1 , ,a h ∈ C, k, h ∈ N,andRex > 0, ζ h,q k,w a 1 , ,a h | s, x 2 h q n 1 ···n h / 0 n 1 , ,n h ≥0 −1 n 1 ···n h q n 1 ···n h w n 1 ···n h a 1 n 1 ··· a h n h x s . 4.10 We note that ζ h,q k,w a 1 , ,a h | s, x is analytic function in the whole complex s-plane. We also see that multipletwisted Barnes’ type q-Euler Zeta functions interpolate multipletwisted Barnes’ type q-Eulerpolynomials at negative integers as follows. Theorem 4.3. For each w ∈ T p , a 1 , ,a h ∈ C, k, h ∈ N,andRe x > 0, ζ h,q k,w a 1 , ,a h |−k, x E h,q k,w a 1 , ,a h | x . 4.11 8 Advances in Difference Equations 5. Multipletwisted Dirichlet’s type q-Eulernumbersandpolynomials Let χ be a Dirichlet’s character with conductor d odd ∈ N and w ∈ T p .Ifwetakefx χxφ w xe tx ,thenwehavef d xfx dχxw d e td w x e tx .From2.2,wederive X χxw x e tx dμ −q x 2 q d−1 i0 −1 d−1−i q i χiw i e ti q d w d e td 1 . 5.1 In view of 5.1, we can define twisted Dirichlet’s type q-Eulernumbers as follows: F q w,χ t 2 q d−1 i0 −1 d−1−i q i χiw i e ti q d w d e td 1 ∞ n0 E q n,χ,w t n n! , t logqw < π d , 5.2 cf. 17, 19, 21, 22.From5.1 and 5.2, we can give Witt’s type formula for twisted Dirichlet’s type q-Eulernumbers as follows. Theorem 5.1. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ T p , n ∈ N ∪{0},wehave X χxw x e tx dμ −q xE q n,χ,w . 5.3 We note that if w 1, then E q n,χ,1 E q n,χ is the generalized q-Eulernumbers attached to χ see 18, 26.From5.2, we also see that F q w,χ t 2 q d−1 i0 −1 d−1−i q i χiw i e ti ∞ l0 q ld w ld e ldt −1 l 2 q ∞ n0 −1 n q n w n χne nt . 5.4 By 5.2 and 5.4,weobtainthat E q k,χ,w d k dt k F q w,χ t | t0 2 q ∞ n0 −1 n q n w n χnn k . 5.5 From 5.5, we can define the l q w,χ -function as follows: l q χ,w s2 q ∞ n0 −1 n q n w n χn n s 5.6 for all s ∈ C.Wenotethatl q χ,w s is analytic function in the whole c omplex s-plane. From 5.5 and 5.6, we can derive the following result. Theorem 5.2. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ T p , n ∈ N ∪{0},wehave l q w,χ −nE q n,χ,w . 5.7 Lee-Chae Jang 9 Now, in view of 5.1, we can define multipletwisted Dirichlet’s type q-Eulernumbers by means of the generating function as follows: F h,q w,χ t 2 q d−1 i0 −1 d−1−i q i χiw i e ti q d w d e td 1 h X χxw x e tx dμ −q x h ∞ n0 E h,q n,χ,w t n n! , 5.8 where |t logqw| <π/d.Wenotethatifw 1, then E q n,χ,1 is a multiple generalized q-Euler number see 22. By using the same method used in 2.8 and 2.9, ∞ n0 X ··· X h-times χ x 1 ··· x h w x 1 ···x h x 1 ··· x h n dμ −q x 1 ···dμ −q x h t n n! ∞ n0 E h,q n,w t n n! . 5.9 From 5.9, we can give Witt’s type formula for multipletwisted Dirichlet’s type q-Euler numbers. Theorem 5.3. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ T p , h ∈ N, and n ∈ N ∪{0},wehave X ··· X h-times χ x 1 ··· x h w x 1 ···x h x 1 ··· x h n dμ −q x 1 ···dμ −q x h E h,q n,χ,w , 5.10 where χx 1 ··· x h χx 1 ···χx h and x 1 ··· x h n l 1 ···l h n l 1 , ,l h ≥0 n! l 1 ! ···l h ! x l 1 1 ···x l h h . 5.11 From 5.10, we also obtain the sums of powers of consecutive multipletwisted Dirichlet’s type q-Eulernumbers as follows. Theorem 5.4. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ T p , h ∈ N, and n ∈ N ∪{0},wehave E h,q k,χ,w l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! E q l 1 ,χ,w ···E q l h ,χ,w . 5.12 Finally, we consider multipletwisted Dirichlet’s type q-Eulerpolynomials defined by means of the generating functions as follows: F q w,χ t, x 2 q d−1 i0 −1 d−1−i q i χiw i e ti q d w d e td 1 h e xt ∞ n0 E h,q n,χ,w x t n n! , 5.13 10 Advances in Difference Equations where |t logqw| <π/dand Rex > 0. From 5.13,wenotethat ∞ n0 X ··· X h-times χ x 1 ···x h w x 1 ···x h x 1 ··· x h x n dμ −q x 1 ···dμ −q x h t n n! ∞ n0 E h,q n,χ,w x t n n! . 5.14 Clearly, we obtain the following two theorems. Theorem 5.5. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ T p , h ∈ N, n ∈ N ∪{0},andRex > 0,wehave X ··· X h-times χ x 1 ··· x h w x 1 ···x h x 1 ··· x h x n dμ −q x 1 ···dμ −q x h E h,q n,χ,w x, 5.15 where x 1 ··· x h x n l 1 ···l h n l 1 , ,l h ≥0 n! l 1 ! ···l h ! x l 1 1 ··· x h x l h . 5.16 Theorem 5.6. Let χ be a Dirichlet’s character with conductor d odd ∈ N. For each w ∈ T p , h ∈ N, n ∈ N ∪{0},andRex > 0,wehave E h,q k,χ,w x l 1 ···l h k l 1 , ,l h ≥0 k! l 1 ! ···l h ! E q l 1 ,χ,w ···E q l h−1 ,χ,w · E q l h ,χ,w x. 5.17 References 1 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. 2 T. 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We also have sums of consecutive multiple twisted q-Euler numbers. . using p-adic q-integrals on Z p ,wedefinemultipletwistedq-Euler numbers and polynomials. We also find Witt’s type formula for multiple twisted q-Euler numbers and discuss some characterizations of multiple. Section 5, we define multiple twisted Dirichlet’s type q-Euler numbers and polynomials and give Witt’s type formula for them. Lee-Chae Jang 3 2. Multiple twisted q-Euler numbers and polynomials In