Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 875098, 11 pages doi:10.1155/2010/875098 Research Article On the Twisted q-Analogs of the Generalized Euler Numbers and Polynomials of H igher Order Lee Chae Jang, 1 Byungje Lee, 2 and Taekyun Kim 3 1 Department of Mathematics and Computer Science, KonKuk University, Chungju 138-701, Republic of Korea 2 Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea 3 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea Correspondence should be addressed to Lee Chae Jang, leechae.jang@kku.ac.kr Received 12 April 2010; Accepted 28 June 2010 Academic Editor: Istvan Gyori Copyright q 2010 Lee Chae J ang 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. We consider the twisted q-extensions of the generalized Euler numbers and polynomials attached to χ. 1. Introduction and Preliminaries Let p be an odd prime number. For n ∈ Z   N ∪{0},letC p n  {ζ | ζ p n  1} be the cyclic group of order p n ,andletT p  lim n →∞ C p n   n≥0 C p n  C p ∞ be the space of locally constant functions in the p-adic number field C p . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C,orp-adic number q ∈ C p .Ifq ∈ C, one normally assumes that |q| < 1. If q ∈ C p , one normally assumes that |1 − q| p < 1. In this paper, we use the notation  x  q  1 − q x 1 − q ,  x  −q  1 −  −q  x 1  q . 1.1 Let d be a fixed positive odd integer. For N ∈ N,weset X  X d  lim ←− N Z dp N Z ,X 1  Z p , 2 Advances in Difference Equations X ∗   0<a<dp a,p1  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 ; compared to 1–16. Let χ be the Dirichlet’s character with an odd conductor d ∈ N. Then the generalized ζ-Euler polynomials attached to χ, E n,χ,ζ x, are defined as F χ,ζ  x, t   2  d−1 l0  −1  l χ  l  ζ l e lt ζ d e dt  1 e xt  ∞  n0 E n,χ,ζ  x  t n n! , for ζ ∈ T p . 1.3 In the special case x  0, E n,χ,ζ  E n,χ,ζ 0 are called the nth ζ-Euler numbers attached to χ. For f ∈ UDZ p ,thep-adic fermionic integral on Z p is defined by I −q  f    Z p f  x  dμ −q  x   lim N →∞ p N −1  x0 f  x  μ −q  x  p N Z p   lim N →∞ p N −1  x0 f  x  −1  x q x  p N  −q , see 7–17. 1.4 Let I −1  lim q → 1  I −q f. Then, we see that  Z p f  x  dμ −1  x    X f  x  dμ −1  x  . 1.5 For n ∈ N,letf n xfx  n. Then, we have  Z p f  x  n  dμ −1  x    −1  n  Z p f  x  dμ −1  x   2 n−1  l0  −1  n−1−l f  l  . 1.6 Thus, we have I −1  f n    −1  n−1 I −1  f   2 n−1  l0  −1  n−1−l f  l  , see 7–17. 1.7 Advances in Difference Equations 3 By 1.7,weseethat  X χ  y  ζ y e xyt dμ −1  y   2  d−1 l0  −1  l χ  l  ζ l e lt ζ d e dt  1 e xt  ∞  n0 E n,χ,ζ  x  t n n! . 1.8 From 1.8, we can derive the Witt’s formula for E n,χ,ζ x as follows:  X χ  x  x n ζ x dμ −1  x   E n,χ,ζ ,  X χ  y  y  x  n ζ y dμ −1  y   E n,χ,ζ  x  , for ζ ∈ T p , see 5–17. 1.9 The nth generalized ζ-Euler polynomials of order k, E k n,χ,ζ , are defined as  2  d−1 l0 ζ l  −1  l χle lt ζ d e dt  1 e xt  k  ∞  n0 E k n,χ,ζ  x  t n n! . 1.10 In the special case x  0, E k n,χ,ζ  E k n,χ,ζ 0 are called the nth ζ-Euler numbers of order k attached to χ. Now, we consider the multivariate p-adic invariant integral on X as follows:  X ···  X  k  i1 χ  x i   e x 1 ···x k xt ζ x 1 ···x k dμ −1  x 1  ···dμ −1  x k    2  d−1 l0  −1  l χle lt ζ d e dt  1  k e xt  ∞  n0 E k n,χ,ζ  x  t n n! . 1.11 By 1.10 and 1.11, we see the Witt’s formula for E k n,χ,ζ x as follows:  X ···  X  k  i1 χ  x i    x 1  ··· x k  x  n ζ x 1 ···x k dμ −1  x 1  ···dμ −1  x k   E k n,χ,ζ  x  . 1.12 The purpose of this paper is to present a systemic study of some formulas of the twisted q-extension of the generalized Euler numbers and polynomials of order k attached to χ. 2. On the Twisted q-Extension of the Generalized Euler Polynomials In this section, we assume that q ∈ C p with |1 − q| p < 1andζ ∈ T p . For d ∈ N with 2  d,let χ be the Dirichlet’s character with conductor d. For h ∈ Z,k ∈ N, let us consider the twisted h, q-extension of the generalized Euler numbers and polynomials of order k attached to χ. 4 Advances in Difference Equations We firstly consider the twisted q-extension of the generalized Euler polynomials of higher order as f ollows: ∞  n0 E n,χ,ζ,q  x  t n n!   X e xy q t ζ y χ  y  dμ −1  y   2 ∞  m0 χ  m  −1  m ζ m e mx q t . 2.1 By 2.1,weseethat  X  x  y  n q χ  y  ζ y dμ −1  y   2 ∞  m0 χ  m  −ζ  m e mx q t  2 d−1  a0 χ  a  −1  a ζ a 1  1 − q  n n  l0  n l   −1  l ζ la q lax 1  q ld ζ ld . 2.2 From the multivariate fermionic p-adic invariant integral on Z p , we can derive the twisted q-extension of the generalized Euler polynomials of order k attached to χ as follows: ∞  n0 E k n,χ,ζ,q  x  t n n!   X ···  X  k  i1 χ  x i   e x 1 ···x k x q t ζ x 1 ···x k dμ −1  x 1  ···dμ −1  x k  . 2.3 Thus, we have E k n,χ,ζ,q  x    X ···  X  k  i1 χ  x i    x 1  ··· x k  x  n q ζ x 1 ···x k dμ −1  x 1  ···dμ −1  x k   d−1  a 1 , ,a k 0  k  i1 χ  a i    −ζ   k j1 a j 2 k  1 − q  n n  l0  n l  −1  l q lx  k j1 a j   1  q ld ζ d   2 k d−1  a 1 , ,a k 0 k  i1  χ  a i    −ζ   k j1 a j ∞  m0  m  k − 1 m  ×  −ζ d  m  x  a 1  ··· a k  md  n q . 2.4 Let F k q,χ,ζ t, x  ∞ n0 E k n,χ,ζ,q xt n /n! be the generating function for E k n,χ,ζ,q x.By2.3, Advances in Difference Equations 5 we easily see that F k q,χ,  t, x   2 k d−1  a 1 , ,a k 0  k  i1 χ  a i    −ζ   k j1 a j ∞  m0  m  k − 1 m  ×  −ζ  dm e xa 1 ···a k md q t  2 k ∞  n 1 , ,n k 0  −ζ  n 1 ···n k  k  i1 χ  n i   e n 1 ···n k x q t . 2.5 Therefore, we obtain the following theorem. Theorem 2.1. For k ∈ N,n ≥ 0, one has E k n,χ,ζ,q  x   2 k ∞  n 1 , ,n k 0  −ζ  n 1 ···n k  k  i1 χ  n i    n 1  ··· n k  x  n q  d−1  a 1 , ,a r 0  k  i1 χ  a i    −ζ   k j1 a j 2 k  1 − q  n n  l0  n l  −1  l q lx  k j1 a j   1  q ld ζ d  n . 2.6 Let h ∈ Z,r ∈ N. Then we define the extension of E r n,χ,ζ,q x as follows: ∞  n0 E h,r n,χ,ζ,q  x  t n n!   X ···  X q  r j1 h−jx j  k  i1 χ  x i   e x  r j1 x j  q t × ζ x 1 ···x r dμ −1  x 1  ···dμ −1  x r  . 2.7 Then, E r n,χ,ζ,q x are called the nth generalized h, q-Euler polynomials of order r attached to χ. In the special case x  0, E r n,χ,ζ,q  E r n,χ,ζ,q 0 are called the nth generalized h, r-Euler numbers of order r.By1.7, we obtain the Witt’s formula f or E r n,χ,ζ,q x as follows: E h,r n,χ,ζ,q  x    X ···  X q  r j1 h−jx j  k  i1 χ  x i   ⎡ ⎣ x  r  j1 x j ⎤ ⎦ n q ζ x 1 ···x r dμ −1  x 1  ···dμ −1  x r   d−1  a 1 , ,a r 0  r  i1 χ  a i    −ζ   r j1 a j q  r j1 a j h−j × 2 r  1 − q  n n  l0  n l  −1  l q lx  r j1 a j   −q dh−rl ζ d ; q d  r , 2.8 where a; q r 1 − a1 − aq ···1 − aq r−1 . 6 Advances in Difference Equations Let  n k  q n q n − 1 q ···n − k  1 q /k q ! n q !/k q !n − k q ! where k q !  k q k − 1 q ···2 q 1 q .From2.8,wenotethat E h,r n,χ,ζ,q  x   2 r  1 − q  n d−1  a 1 , ,a r 0  r  i1 χ  a i    −ζ   r j1 a j q  r j1 h−ja j × n  l0  n l   −1  l q lx  r j1 a j  ∞  m0  m  r − 1 m  q d  −ζ d  m q dh−rm q ldm  2 r d−1  a 1 , ,a r 0  r−1  i1 χ  a i    −ζ   r j1 a j q  r j1 h−ja j × ∞  m0  m  r − 1 m  q d  −ζ d  m q dh−rm 1  1 − q  n  1 − q dmx  k j1 a j /d  n  2 r  d  n q ∞  m0  m  r − 1 m  q d  −ζ d  m q dh−rm d−1  a 1 , ,a r 0  r−1  i1 χ  a i   ×  −ζ   r j1 a j q  r j1 h−ja j ⎡ ⎣ m  x   d−1 j1 a j d ⎤ ⎦ n q d . 2.9 Let F h,r q,χ,ζ t, x  ∞ n0 E h,r n,χ,ζ,q xt n /n! be the generating function for E h,r n,χ,ζ,q x.From 2.8, we can easily derive F h,r q,χ,ζ  t, x   2 r ∞  n 1 , ,n r 0 q  r j1 h−jn j  −ζ   r j1 n j ⎛ ⎝ r  j1 χ  n j  ⎞ ⎠ e n 1 ···n r x q t  2 r ∞  m0  m  r − 1 m  q  −ζ d  m q dh−rm d−1  a 1 , ,a r 0  r−1  i1 χ  a i   ×  −ζ   r j1 a j q  r j1 h−ja j e mdx  r j1 a j  q t . 2.10 By 2.10, we obtain the following theorem. Advances in Difference Equations 7 Theorem 2.2. For h ∈ Z, r ∈ N, one has E h,r n,χ,ζ,q  x   2 r ∞  n 1 , ,n r 0 q  r j1 h−jn j  −ζ   r j1 n j ⎛ ⎝ r  j1 χ  n j  ⎞ ⎠  n 1  ··· n r  x  n q  2 r  d  n q ∞  m0  m  r − 1 m  q  −ζ d  m q dh−rm d−1  a 1 , ,a r 0  r−1  i1 χ  a i   ×  −ζ   r j1 a j q  r j1 h−ja j  m  x   r j1 a j d  n q d  d−1  a 1 ···a r 0 ⎛ ⎝ r  j1 χ  n j  ⎞ ⎠  −ζ   r j1 a j q  r j1 h−ja j × 2 r  1 − q  n n  l0  n l  −1  l q lx  r j1 a j   −q dh−rl ζ d ; q d  r . 2.11 Let h  r. Then we see that E r,r n,χ,ζ,q  x   2 r  1 − q  n d−1  a 1 , ,a r 0  r−1  i1 χ  a i    −ζ   r i1 a i q  r j1 h−ja j × n  l0  n l  −1  l q l  r j1 a j x  −q ld ζ d ; q d  r  2 r  d  n q ∞  m0  m  r − 1 m  q  −ζ  m d−1  a 1 , ,a r 0  r  i1 χ  a i   ×  −ζ   r j1 a j q  r j1 r−ja j  m  x   r j1 a j d  n q d . 2.12 It is easy to see that  X ···  X  k  i1 χ  x i   q  r j1 h−jx j xm ζ x 1 ···x r dμ −1  x 1  ···dμ −1  x r   d−1  a 1 , ,a r 0  r  i1 χ  a i   q mx  r j1 h−ja j  −ζ   r j1 a j ×  X ···  X q  r j1 m−jx j dμ −1  x 1  ···dμ −1  x r   2 r q mx  d−1 a 1 , ,a r 0   r j1 χ  a j   q  r j1 m−ja j  −ζ   r j1 a j  −q dm−r ζ d ; q d  r . 2.13 8 Advances in Difference Equations Thus, we have 2 r q mx  d−1 a 1 , ,a r 0   r j1 χ  a j   q  r j1 m−ja j  −ζ   r j1 a j  −q dm−r ζ d ; q d  r   X ···  X   x  x 1  ··· x r  q  q − 1   1  m q −  r j1 jx j ζ x 1 ···x r × ⎛ ⎝ r  j1 χ  x j  ⎞ ⎠ dμ −1  x 1  ···dμ −1  x r   m  l0  m l   q − 1  l  X ···  X ⎛ ⎝ r  j1 χ  x j  ⎞ ⎠ ×  x  x 1  ··· x r  l q q −  r j1 jx j ζ x 1 ···x r dμ −1  x 1  ···dμ −1  x r   m  l0  m l   q − 1  l E 0,r l,χ,ζ,q  x  . 2.14 By 2.14, we obtain the following theorem. Theorem 2.3. For d, k ∈ N with 2  d, one has 2 r q mx  d−1 a 1 , ,a r 0   r j1 χ  a j   q  r j1 m−ja j  −ζ   r j1 a j  −q dm−r ζ d ; q d  r  m  l0  m l   q − 1  l E 0,r l,χ,ζ,q  x  . 2.15 By 1.7,weeasilyseethat  X f  x  d  dμ −1  x    X f  x  dμ −1  x   2 d−1  l0  −1  l f  l  . 2.16 Thus,we have q dh−1  X ···  X  x  d  x 1  ··· x r  n q q  r j1 r−jx j ζ  r j1 x j × ⎛ ⎝ r  j1 χ  x j  ⎞ ⎠ dμ −1  x 1  ···dμ −1  x r   −  X ···  X  x  x 1  ··· x r  n q q  r j1 r−jx j ζ  r j1 x j × ⎛ ⎝ r  j1 χ  x j  ⎞ ⎠ dμ −1  x 1  ···dμ −1  x r   2 d−1  l0 χ  l  −ζ  l  X ···  X  x  l  x 2  ··· x r  n q ⎛ ⎝ r−1  j1 χ  x j1  ⎞ ⎠ × q  r−1 j1 x j1 h−1−j ζ x 2 x 3 ···x r dμ −1  x 2  ···dμ −1  x r  . 2.17 By 2.17, we obtain the following theorem. Advances in Difference Equations 9 Theorem 2.4. For h ∈ Z,d ∈ N with 2  d, one has q dh−1 E h,r n,χ,ζ,q  x  d   E h,r n,χ,ζ,q  x   2 d−1  l0 χ  l  −1  l E h−1,r−1 n,χ,ζ,q  x  l  . 2.18 It is easy to see that q x E h1,r n,χ,ζ,q  x    q − 1  E h,r n1,χ,ζ,q  E h,r n,χ,ζ,q  x  . 2.19 Let F h,1 q,χ,ζ t, x  ∞ n0 E h,1 n,χ,ζ,q xt n /n!. Then we note that F h,1 q,χ,ζ  t, x   2 ∞  n0 χ  n  q h−1n  −ζ  n e nx q t . 2.20 From 2.20, we can derive E h,1 n,χ,ζ,q  x   2 ∞  m0 χ  m  q h−1m  −ζ  m  m  x  n q  2  1 − q  n d−1  a0 χ  a  −ζ  a n  l0  n l  −1  l q lxa  1  q ld ζ d  . 2.21 3. Further Remark In this section, we assume that q ∈ C with |q| < 1. Let χ be the Dirichlet’s character with an odd conductor d ∈ N. From the Mellin transformation of F h,r q,χ,ζ t, x in 2.10,wenotethat 1 Γ  s   F h,r q,χ,ζ  −t, x  t s−1 dt  2 r ∞  m 1 , ,m r 0 q  r j1 h−jm j  −ζ  m 1 ···m r   r j1 χ  m j    m 1  ··· m r  x  s q , 3.1 where h, s ∈ C, x /  0, −1, −2, ,and r ∈ N, ζ  e 2πi/d .By3.1, we can define the Dirichlet’s type multiple h, q-l-function as follows. Definition 3.1. For s ∈ C, x ∈ R with x /  0, −1, −2, , one defines the Dirichlet’s type multiple h, q-l-function related to higher order h, q-Euler polynomials as l h,r q  s, x | χ   2 r ∞  m 1 ,··· ,m r 0 q  r j1 h−jm j  −ζ  m 1 ···m r   r i1 χ  m i    m 1  ··· m r  x  s q , 3.2 where s, h ∈ C, x /  0, −1, −2, ···, r ∈ N,andζ  e 2πi/d . 10 Advances in Difference Equations Note that l h,r q s, x | χ is analytic continuation in whole complex s-plane. In 2.10,we note that F h,r q,χ,ζ  t, x   2 r ∞  n 1 , ,n r 0 q  r j1 h−jn j  −ζ  n 1 ···n r ⎛ ⎝ r  j1 χ  n j  ⎞ ⎠ e n 1 ···n r x q t  ∞  n0 E h,r n,χ,ζ,q  x  t n n! . 3.3 By Laurent series and Cauchy residue theorem in 3.1 and 3.3,weobtainthe following theorem. Theorem 3.2. Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ  e 2πi/d . For h, s ∈ C, x /  0, −1, −2, ,r ∈ N, and n ∈ Z  , one has l h,r q  −h, x | χ   E h,r n,χ,ζ,q  x  . 3.4 References 1 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. 2 T. Kim, “On a q-analogue of the p-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999. 3 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on Z p ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. 4 T. Kim, “Note on the Euler q-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009. 5 T. Kim, “Note on multiple q-zeta functions,” to appear in Russian Journal of Mathematical Physics. 6 T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A, vol. 43, no. 25, Article ID 255201, 11 pages, 2010. 7 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. 8 T. Kim, “On the multiple q-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 481–486, 2008. 9 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008. 10 T. Kim, “Note on the q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008. 11 T. Kim, “Note on Dedekind type DC sums,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 249–260, 2009. 12 T. Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003. 13 S H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009. 14 L C. Jang, “A study on the distribution of twisted q-Genocchi polynomials,” Advanced Studies in Contemporary Mathematics , vol. 19, no. 2, pp. 181–189, 2009. 15 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order w-q-Genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009. [...]... in Difference Equations 11 16 M Can, M Cenkci, V Kurt, and Y Simsek, Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius -Euler -functions,” Advanced Studies in Contemporary Mathematics, vol 18, no 2, pp 135–160, 2009 17 Y.-H Kim, W Kim, and C S Ryoo, On the twisted q -Euler zeta function associated with twisted q -Euler numbers, ” Proceedings of the Jangjeon Mathematical Society, . E k n,χ,ζ  x  . 1.12 The purpose of this paper is to present a systemic study of some formulas of the twisted q-extension of the generalized Euler numbers and polynomials of order k attached to χ. 2. On the Twisted. consider the twisted h, q-extension of the generalized Euler numbers and polynomials of order k attached to χ. 4 Advances in Difference Equations We firstly consider the twisted q-extension of. Corporation Advances in Difference Equations Volume 2010, Article ID 875098, 11 pages doi:10.1155/2010/875098 Research Article On the Twisted q-Analogs of the Generalized Euler Numbers and Polynomials