Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 579509, 6 pages doi:10.1155/2010/579509 Research Article Multivariate Twisted p-Adic q-Integral on Z p Associated with Twisted q-Bernoulli Polynomials and Numbers Seog-Hoon Rim, Eun-Jung Moon, Sun-Jung Lee, and Jeong-Hee Jin Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea Correspondence should be addressed to Seog-Hoon Rim, shrim@knu.ac.kr Received 19 June 2010; Accepted 2 October 2010 Academic Editor: Ulrich Abel Copyright q 2010 Seog-Hoon Rim 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. Recently, many authors have studied twisted q-Bernoulli polynomials by using the p-adic invariant q-integral on Z p . In this paper, we define the twisted p-adic q-integral on Z p and extend our result to the twisted q-Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted q-Bernoulli polynomials. 1. Introduction Let p be a fixed prime number. Throughout this paper, the symbols Z, Z p , Q p , C, and C p will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Q p , respectively. Let N be the set of natural numbers and Z   N ∪{0}. Let v p be the normalized exponential valuation of C p with |p| p  p −v p p  1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or p-adic number q ∈ C p . If q ∈ C, one normally assumes that |q| < 1. If q ∈ C p , then we assume that |q − 1| p < 1. For n ∈ N, let T p be the p-adic locally constant space defined by T p   n≥1 C p n  lim n →∞ C p n  C p ∞ , 1.1 where C p n  {ζ ∈ C p | ζ p n  1 for some n ≥ 0} is the cyclic group of order p n . 2 Journal of Inequalities and Applications Let UDZ p  be the space of uniformly differentiable function on Z p . For f ∈ UDZ p , the p-adic invariant q-integral on Z p is defined as I q  f    Z p f  x  dμ q  x   lim N →∞ 1  p N  q p N −1  x0 f  x  q x , 1.2 compare with 1–3. It is well known that the twisted q-Bernoulli polynomials of order k are defined as e xt  t e t ζq − 1  k  ∞  n0 β k n,ζ,q  x  t n n! ,ζ∈ T p , 1.3 and β k n,ζ,q  β k n,ζ,q 0 are called the twisted q-Bernoulli numbers of order k. When k  1, the polynomials and numbers are called the twisted q-Bernoulli polynomials and numbers, respectively. When k  1andq  1, the polynomials and numbers are called the twisted Bernoulli polynomials and numbers, respectively. When k  1,q  1, and ζ  1, the polynomials and numbers are called the ordinary Bernoulli polynomials and numbers, respectively. Many authors have studied the twisted q-Bernoulli polynomials by using the properties of the p-adic invariant q-integral on Z p cf. 4. In this paper, we define the twisted p-adic q-integral on Z p and extend our result to the twisted q-Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted q-Bernoulli polynomials. 2. Multivariate Twisted p-Adic q-Integral on Z p Associated with Tw is t ed q-Bernoulli Polynomials In this section, we assume that q ∈ C p with |q−1| p < 1. For ζ ∈ T p , we define the q, ζ-numbers as  k  q,ζ  1 − q k ζ 1 − q , for k ∈ Z p . 2.1 Note that k q k q,1 1 − q k /1 − q. Let us define  n k  q,ζ   n  q,ζ !  k  q,ζ !  n − k  q,ζ ! , 2.2 where k q,ζ ! k q,ζ k − 1 q,ζ ···1 q,ζ . Note that  n k    n k  1,1  n!/k!n − k!. Journal of Inequalities and Applications 3 Now we construct the twisted p-adic q-integral on Z p as follows: I q,ζ  f    Z p f  x  dμ q,ζ  x   lim N →∞ p N −1  x0 f  x  μ q,ζ  x  p N Z p   lim N →∞ 1  p N  q p N −1  x0 f  x  q x ζ x , 2.3 where μ q,ζ x  p N Z p q x ζ x /p N  q . From the definition of the twisted p-adic q-integral on Z p , we can consider the twisted q-Bernoulli polynomials and numbers of order k as follows: β k n,q,ζ  x    Z k p  x 1  x 2  ··· x k  x  n q dμ q,ζ  x 1  dμ q,ζ  x 2  ···dμ q,ζ  x k   lim N →∞ 1  p N  k q p N −1  x 1 , ,x k 0  x 1  x 2  ··· x k  x  n q q x 1 x 2 ···x k ζ x 1 x 2 ···x k  1  1 − q  n n  l0  n l   −1  l q lx lim N →∞ 1  p N  k q p N −1  x 1 , ,x k 0 q l1x 1 ···l1x k ζ x 1 ···x k  1  1 − q  n n  l0  n l   −1  l q lx  l  1  k  l  1  k q,ζ . 2.4 In the special case x  0in2.4, β k n,q,ζ 0β k n,q,ζ are called the twisted q-Bernoulli numbers of order k. If we take k  1andζ  1in2.4, we can easily see that β n,q  x   1  1 − q  n n  l0  n l   −1  l q lx l  1  l  1  q . 2.5 compare with 4. Theorem 2.1. For k ∈ Z  and ζ ∈ T p , we have β k n,q,ζ  x   1  1 − q  n n  l0  n l   −1  l q lx  l  1  k  l  1  k q,ζ . 2.6 4 Journal of Inequalities and Applications Moreover, if we take x  0 in Theorem 2.1, then we have the following identity for the twisted q-Bernoull numbers β k n,q,ζ  1  1 − q  n n  l0  n l   −1  l  l  1  k  l  1  k q,ζ . 2.7 From the definition of multivariate twisted p-adic q-integral, we also see that β k n,q,ζ  x    Z k p  x 1  x 2  ··· x k  x  n q dμ q,ζ  x 1  dμ q,ζ  x 2  ···dμ q,ζ  x k   n  l0  n l  q lx  x  n−l q  Z k p  x 1  x 2  ··· x k  l q dμ q,ζ  x 1  dμ q,ζ  x 2  ···dμ q,ζ  x k   n  l0  n l  q lx  x  n−l q β k l,q,ζ . 2.8 Corollary 2.2. For k ∈ Z  and ζ ∈ T p , one obtains β k n,q,ζ  x   n  l0  n l  q lx  x  n−l q β k l,q,ζ . 2.9 Note that q nx 1 ···x k   n  l0  n l   q − 1  l  x 1  ··· x k  l q . 2.10 We have  Z k p q nx 1 ···x k  dμ q,ζ  x 1  dμ q,ζ  x 2  ···dμ q,ζ  x k   n  l0  n l   q − 1  l β k l,q,ζ . 2.11 It is easy to see that  Z k p q nx 1 ···x k  dμ q,ζ  x 1  dμ q,ζ  x 2  ···dμ q,ζ  x k   lim N →∞ 1  p N  k q p N −1  x 1 , ,x k 0 q nx 1 ···x k  q x 1 ···x k ζ x 1 ···x k   n  1  k  n  1  k q,ζ . 2.12 By 2.11 and 2.12, we obtain the following theorem. Journal of Inequalities and Applications 5 Theorem 2.3. For n ∈ Z  ,k∈ N and ζ ∈ T p , one has n  l0  n l   q − 1  l β k l,q,ζ   n  1  k  n  1  k q,ζ . 2.13 Now we consider the modified extension of the twisted q-Bernoulli polynomials of order k as follows: B k n,q,ζ  x   1  1 − q  n n  i0  −1  i  n i  q ix  Z k p q  k l1 k−lix i dμ q,ζ  x 1  ···dμ q,ζ  x k  . 2.14 In the special case x  0, we write B k n,q,ζ  B k n,q,ζ 0, which are called the modified extension of the twisted q-Bernoulli numbers of order k. From 2.14, we derive that B k n,q,ζ  x   1  1 − q  n n  i0  −1  i  n i   i  k  ···  i  1   i  k  q,ζ ···  i  1  q,ζ q ix  1  1 − q  n n  i0  −1  i  n i   ik k  k!  ik k  q,ζ  k  q,ζ ! q ix . 2.15 Therefore, we obtain the following theorem. Theorem 2.4. For n ∈ Z  ,k∈ N and ζ ∈ T p , one has B k n,q,ζ  x   1  1 − q  n n  i0  −1  i  n i   ik k  k!  ik k  q,ζ  k  q,ζ ! q ix . 2.16 Now, we define B −k n,q,ζ x as follows: B −k n,q,ζ  x   1  1 − q  n n  i0  −1  i  n i  q ix  Z k p q  k l1 k−lix i dμ q,ζ  x 1  ···dμ q,ζ  x k  . 2.17 By 2.17, we can see that B −k n,q,ζ  x   1  1 − q  n n  i0  −1  i  n i   ik k  q,ζ  k  q,ζ !  ik k  k! q ix . 2.18 Therefore, we obtain the following theorem. 6 Journal of Inequalities and Applications Theorem 2.5. For n ∈ Z  ,k∈ N and ζ ∈ T p , one has B −k n,q,ζ  x   1  1 − q  n n  i0  −1  i  i  k k  q,ζ  nk n−i   k  q,ζ !  nk k  k! q ix . 2.19 In 2.19, we can see the relations between the binomial coefficients and the modified extension of the twisted q-Bernoulli polynomials of order k. Acknowledgments The authors would like to thank the anonymous referee for his/her excellent detail comments and suggestions. This Research was supported by Kyungpook National University Research Fund, 2010. References 1 L C. Jang, “Multiple twisted q-Euler numbers and polynomials associated with p-adic q-integrals,” Advances in Difference Equations, Article ID 738603, 11 pages, 2008. 2 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002. 3 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. 4 T. Kim, “Sums of products of q-Bernoulli numbers,” Archiv der Mathematik, vol. 76, no. 3, pp. 190–195, 2001. . Corporation Journal of Inequalities and Applications Volume 2010, Article ID 579509, 6 pages doi:10.1155/2010/579509 Research Article Multivariate Twisted p-Adic q-Integral on Z p Associated with Twisted. studied twisted q-Bernoulli polynomials by using the p-adic invariant q-integral on Z p . In this paper, we define the twisted p-adic q-integral on Z p and extend our result to the twisted q-Bernoulli. the twisted q-Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted q-Bernoulli polynomials. 2. Multivariate Twisted p-Adic q-Integral on Z p Associated