Báo cáo hóa học: " Research Article Congruences for Generalized q-Bernoulli Polynomials" pot

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 270713, 19 pages doi:10.1155/2008/270713 Research Article Congruences for Generalized q-Bernoulli Polynomials Mehmet Cenkci and Veli Kurt Department of Mathematics, Akdeniz University, 07058 Antalya, Turkey Correspondence should be addressed to Mehmet Cenkci, cenkci@akdeniz.edu.tr Received 9 December 2007; Accepted 15 February 2008 Recommended by Andrea Laforgia In this paper, we give some further properties of p-adic q-L-function of two variables, which is recently constructed by Kim 2005 and Cenkci 2006. One of the applications of these proper- ties yields general classes of congruences for generalized q-Bernoulli polynomials, which are q- extensions of the classes for generalized Bernoulli numbers and polynomials given by Fox 2000, Gunaratne 1995,andYoung1999, 2001. Copyright q 2008 M. Cenkci and V. Kurt. 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 and primary concepts For n ∈ Z, n ≥ 0, Bernoulli numbers B n originally arise in the study of finite sums of a given power of consecutive integers. They are given by B 0  1, B 1  −1/2, B 2  1/6, B 3  0, B 4  −1/30, ,withB 2n1  0 for odd n>1, and B n  − 1 n  1 n−1  m0  n  1 m  B m , 1.1 for all n ≥ 1. In the symbolic notation, Bernoulli numbers are given recursively by B  1 n − B n  δ n,1 , 1.2 with the usual convention about replacing B j by B j ,whereδ n,1 is the Kronecker symbol. The Bernoulli polynomials B n z can be expressed in the form B n zB  z n  n  m0  n m  B m z n−m , 1.3 2 Journal of Inequalities and Applications for an indeterminate z. The generating functions of these numbers and polynomials are given, respectively, by Ft t e t − 1  ∞  n0 B n t n n! , Fz, t t e t − 1 e zt  ∞  n0 B n z t n n! , 1.4 for |t| < 2 π. One of the notable facts about Bernoulli numbers and polynomials is the relation between the Riemann and the Hurwitz or generalized zeta functions. Theorem 1.1 see 1, 2. For every integer n ≥ 1, ζ1 − n− B n n ,ζ1 − n, z− B n z n , 1.5 where ζs and ζs, z are the Riemann and the Hurwitz (or generalized) zeta functions, defined, re- spectively, by ζs ∞  m1 1 m s ,ζs, z ∞  m0 1 m  z s , 1.6 with s ∈ C, s > 1,andz ∈ C with z > 0. Among various generalizations of Bernoulli numbers and polynomials, generalization with a primitive Dirichlet character χ has a special case of attention. Definition 1.2 see 2, 3. For a primitive Dirichlet character χ having conductor f ∈ Z, f ≥ 1, the generalized Bernoulli numbers B n,χ and polynomials B n,χ z associated with χ are defined by F χ t f  a1 χate at e ft − 1  ∞  n0 B n,χ t n n! , F χ z, t f  a1 χate azt e ft − 1  ∞  n0 B n,χ z t n n! , 1.7 respectively, for |t| < 2 π/f. When χ  1, the classical Bernoulli numbers and polynomials are obtained in that B n,1  −1 n B n and B n,1 z−1 n B n −z. The generalized Bernoulli numbers and polynomials can be expressed in terms of Bernoulli polynomials as B n,χ  f n−1 f  a1 χaB n  a f  , B n,χ zf n−1 f  a1 χaB n  a  z f  . 1.8 M. Cenkci and V. Kurt 3 Given a primitive Dirichlet character χ having conductor f, the Dirichlet L-function as- sociated with χ is defined by 1, 2 Ls, χ ∞  m1 χm m s , 1.9 where s ∈ C,Res > 1. It is well known 2 that Ls, χ may be analytically continued to the whole complex plane, except for a simple pole at s  1whenχ  1, in which case it reduces to Riemann zeta function, ζsLs, 1. The generalized Bernoulli numbers share a particular relationship with the Dirichlet L-function in that L1 − n, χ− B n,χ n , 1.10 for n ∈ Z, n ≥ 1. Let p be a fixed prime number. Throughout this paper, Z p , Q p , C,andC p will, respec- tively, denote 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 .Let|·| p denote the p-adic ab- solute value on Q p , normalized so that |p| p  p −1 .Letp ∗  4ifp  2andp ∗  p otherwise. Note that there exist φp ∗  distinct solutions, modulo p ∗ , to the equation x φp ∗  − 1  0, and each solution must be congruent to one of the values a ∈ Z,where1≤ a ≤ p ∗ − 1, a, p1. Thus, by Hensel’s lemma, given a ∈ Z with a, p1, there exists a unique wa ∈ Z p such that wa ≡ amod p ∗ Z p . Letting wa0fora ∈ Z such that a, p /  1, it can be seen that w is actually a Dirichlet character having conductor f w  p ∗ , called the Teichm ¨ uller character. Let x  wxx.Thenx≡1mod p ∗ Z p . In the sense of product of characters, let χ n  χw −n . This implies that f χ n | fp ∗ . Since χ  χ n w n , f | f χ n p ∗ is also true. Thus, f and f χ n differ by a factor that is a power of p. During the development of p-adic analysis, researches were made to derive a meromor- phic function, defined over the p-adic number field, that would interpolate the same, or at least similar values as the Dirichlet L-function at nonpositive integers. In 4, Kubota and Leopoldt proved the existence of such a function, considered as p-adic equivalent of the Dirichlet L- function. Proposition 1.3 see 3, 4. There exists a unique p-adic meromorphic (analytic if χ /  1) function L p s, χ, s ∈ Z p ,forwhich L p 1 − n, χ  1 − χ n pp n−1  L  1 − n, χ n  , 1.11 for n ∈ Z, n ≥ 1. By 1.10, this function yields the values L p 1 − n, χ− 1 n  1 − χ n pp n−1  B n,χ n , 1.12 for n ∈ Z, n ≥ 1. Since the time of the work of Kubota and Leopoldt, many mathematicians have derived the existence and generalizations of L p s, χ by various means 5–12. In particular, Washington 11 derived the function by elementary means and expressed it in an explicit form. Let D denote the region D   s ∈ C p : |s − 1| p < |p| 1/p−1 p |p ∗ | −1 p  . 1.13 4 Journal of Inequalities and Applications Theorem 1.4 see 11. Let F be a positive integer multiple of p ∗ and f, and let L p s, χ 1 s − 1 1 F F  a1 a,p1 χaa 1−s ∞  m0  1 − s m   F a  m B m . 1.14 Then, L p s, χ is analytic for s ∈ D,whenχ /  1, and meromorphic for s ∈ D, with a simple pole at s  1, having residue 1 − 1/p,whenχ  1. Furthermore, for each n ∈ Z, n ≥ 1, L p 1 − n, χ− 1 n  1 − χ n pp n−1  B n,χ n . 1.15 Thus, L p s, χ vanishes identically if χ−1−1. In 6, Fox derived a p-adic function L p s, z, χ,wherez ∈ C p , |z| p ≤ 1, and s ∈ D,that interpolates the values L p 1 − n, z, χ− 1 n  B n,χ n  p ∗ z  − χ n pp n−1 B n,χ n  p −1 p ∗ z  , 1.16 for positive integers n. By applying the method that Washington used to derive Theorem 1.4, Fox 7 obtained L p s, z, χ by elementary means and expressed it in an explicit form. Theorem 1.5 see 7. Let F be a positive integer multiple of p ∗ and f, and let L p s, z, χ 1 s − 1 χ−1 F F  a1 a,p1 χa  a − p ∗ z  1−s × ∞  m0  1 − s m   F a − p ∗ z  m B m . 1.17 Then, L p s, z, χ is analytic for z ∈ C p , |z| p ≤ 1, provided that s ∈ D, except for s /  1 when χ  1. Also, if z ∈ C p , |z| p ≤ 1, this function is analytic for s ∈ D when χ /  1, and meromorphic for s ∈ D, withasimplepoleats  1, having residue 1 − 1/p,whenχ  1. Furthermore, for each n ∈ Z, n ≥ 1, L p 1 − n, z, χ− 1 n  B n,χ n  p ∗ z  − χ n pp n−1 B n,χ n  p −1 p ∗ z  . 1.18 In 12, Young gave p-adic integral representations for the two-variable p-adic L-function introduced by Fox. These representations leaded to generalizations of some formulas of Dia- mond 13, 14 and of Ferrero and Greenberg 15 for p-adic L-functions in terms of the p-adic gamma and log gamma functions. But, his work was restricted to character χ such that the conductor of χ 1 is not a power of p. The explicit formula given in Theorem 1.5 by Fox yielded to derive formulas similar to that obtained by Young, but for all primitive Dirichlet character χ. In 16, Carlitz defined q-extensions of Bernoulli numbers and polynomials, and proved properties generalizing those satisfied by B n and B n z. When talking about q-extensions, q can be considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ C p .If q ∈ C, then it is assumed that |q| < 1andifq ∈ C p , then it is assumed that |1 − q| p <p −1/p−1 ,so M. Cenkci and V. Kurt 5 that q x  exp log p q for |x| p ≤ 1, where log is the Iwasawa p-adic logarithm function see 3, Chapter 4. The q-Bernoulli numbers β n,q , n ∈ Z, n ≥ 0, are usually defined by β 0,q  q − 1 log q ,  qβ q  1  n − β n,q  δ n,1 , 1.19 where the usual convention about replacing β j q by β j,q in the binomial expansion is understood 8, 17–24. It follows from 1.19 that β n,q  1 1 − q n n  i0  n i  −1 i i i q , 1.20 where it is understood that for i  0, the function i/i q  1. We use the notation x q  1 − q x 1 − q , 1.21 so that lim q→1 x q  x for any x ∈ C in the complex case and x ∈ C p with |x| p ≤ 1inthep-adic case. In 8, 9, Kim defined q-Bernoulli polynomials β n,q z, n ∈ Z, n ≥ 0, as β n,q z  q z β q  z  n  n  m0  n m  q mz β m,q z n−m q  1 1 − q n n  i0  n i  −1 i q iz i i q . 1.22 Some basic properties of q-Bernoulli polynomials β n,q z similar to those of Bernoulli polyno- mials B n z can be deduced from 1.22see also 25. For instance, we have β n,q −1 1 − z−1 n q n−1 β n,q z, 1.23 β n,q 1  z − β n,q znq z z n−1 q , 1.24 β n,q z  τ n  m0  n m  q mz β m,q τz n−m q . 1.25 Let χ be a Dirichlet character with conductor f. The generalized q-Bernoulli polynomials associated with χ, β n,q,χ z, n ∈ Z, n ≥ 0, are defined by 8, 9 β n,q,χ zf n−1 q f  a1 χaβ n,q f  a  z f  . 1.26 For z  0, β n,q,χ 0β n,q,χ are the generalized q-Bernoulli numbers, β n,q,χ f n−1 q f  a1 χaβ n,q f  a f  . 1.27 6 Journal of Inequalities and Applications From 1.25, 1.26,and1.27, β n,q,χ z n  m0  n m  q mz β m,q,χ z n−m q . 1.28 An important property that the polynomials β n,q,χ z satisfy is the following, which can be proved by using 1.24 and 1.26: Proposition 1.6. For m ∈ Z, m ≥ 1, β n,q,χ mf  z − β n,q,χ zn mf  a1 χaq az a  z n−1 q 1.29 for all n ∈ Z, n ≥ 1. Note that for χ  1 i.e., f  1, z  0, and q → 1, Proposition 1.6 reduces to m  a1 a n−1  1 n  B n,1 m − B n,1  , 1.30 which is the well-known property of Bernoulli numbers and polynomials. Let K be an extension of Q p contained in C p . An infinite series  a n ,a n ∈ K,converges in K if and only if |a n | p → 0, as n →∞.LetKx and Kx be, respectively, the algebras of formal power series and of polynomials in x. Then, Ax  a n x n ∈ Kx converges at x  η, η ∈ C p , if and only if |a n η n | p → 0, as n →∞. The following is a uniqueness property for power series found in 3. Lemma 1.7. Let Ax,Bx ∈ Kx such that each converges in a neighborhood of 0 in C p .If Aη n Bη n  for a sequence {η n }, η n /  0,in C p such that η n → 0,thenAxBx. Any positive integer n can be uniquely expressed in the form n  k  m0 a m p m , 1.31 where a m ∈ Z,0≤ a m ≤ p − 1, for m  0, 1, ,kand a k /  0. For such n,let s p n k  m0 a m 1.32 be the sum of the p-adic digits of n with s p 00. For any n ∈ Z,letv p n be the highest power of p dividing n. The function v p is additive and relates s p by means of v p n! n − s p n p − 1 1.33 for all n ≥ 0. For n ≥ 1, 1.33 implies that v p n! ≤ n − 1 p − 1 . 1.34 M. Cenkci and V. Kurt 7 We denote a particular subring of C p as o   a ∈ C p : |a| p < 1  . 1.35 If z ∈ C p such that |z| p ≤|p| m p ,wherem ∈ Q,thenz ∈ p m o, and this can be also written as z ≡ 0mod p m o. Let the set R be defined as R   a ∈ C p : |a| p <p −1/p−1  . 1.36 Obviously, R ⊂ o. Since |1 − q| p <p −1/p−1 for q ∈ C p ,wehave1− q ∈ R, which implies that q ≡ 1mod R.Leta : q a q w −1 a. For the context in the sequel, an extension of a : q is needed. Since w can be considered as a Dirichlet character of conductor p ∗ , wa  p ∗ zwa for a ∈ Z with a, p1. Thus, a  p ∗ z : q can be defined by  a  p ∗ z : q    a  p ∗ z  q wa . 1.37 If z ∈ C p such that |z| p ≤ 1, then for any a ∈ Z,  a  p ∗ z  q a q  q a  p ∗ z  q ≡ a q mod R. 1.38 Thus, a  p ∗ z : q≡1mod p ∗ R. Let F be a positive integer multiple of f and p ∗ .In9, Kim defined p-adic q-L-function of two variables L p,q s, z, χ as follows: L p,q s, z, χ 1 s − 1 1 F q F  a1 a,p1 χa  a  p ∗ z : q  1−s × ∞  m0  1 − s m  β m,q F q ap ∗ zm  F a  p ∗ z  m q ap ∗ z . 1.39 The analytic properties of L p,q s, z, χ are given by the following theorem. Theorem 1.8 see 9. Let F be a positive multiple of f and p ∗ and let L p,q s, z, χ be as in 1.39. Then, L p,q s, z, χ is analytic for z ∈ C p , |z| p ≤ 1, provided that s ∈ D, except for s  1 if χ /  1. Moreover, if z ∈ C p , |z| p ≤ 1, then this function is analytic for s ∈ D if χ /  1 and meromorphic for s ∈ D with a simple pole at s  1 with residue 1/F q q F − 1/ log q1 − 1/p if χ  1. Furthermore, for n ∈ Z, n ≥ 1, L p,q 1 − n, z, χ− 1 n  β n,q,χ n  p ∗ z  − χ n pp n−1 q β n,q p ,χ n  p −1 p ∗ z   . 1.40 Kim 9 also gave a p-adic integral representation for the function L p,q s, z, χ and derived a q-extension of the generalized Diamond-Ferrero-Greenberg formula for the two- variable p-adic L-function in terms of p-adic gamma and log-gamma functions. In 5, first author derived L p,q s, z, χ by using convergent power series, a method developed by Iwasawa 3. Resulting function from this derivation is in closed form but satisfies same properties of the function defined by 1.39. The main motivation of this paper is to derive general classes of congruences for gener- alized q-Bernoulli polynomials by making use of the function L p,q s, z, χ. These classes are ob- tained as an application of the difference formula see 2.12 for the p-adic q-L-function of two 8 Journal of Inequalities and Applications variables, which generalizes Proposition 1.6 and thus the well-known formula for Bernoulli numbers and polynomials 1.30. 2. Properties of L p,q s, z, χ Recall that L p,q s, z, χ, z ∈ C p , |z| p ≤ 1, interpolates the values L p,q 1 − n, z, χ− 1 n b n z, q, χ, 2.1 for n ∈ Z, n ≥ 1, where b n z, q, χβ n,q,χ n  p ∗ z  − χ n pp n−1 q β n,q p ,χ n  p −1 p ∗ z  . 2.2 Lemma 2.1. For all n ∈ Z, n ≥ 1, b n  − z, q −1 ,χ   χ−1q n−1 b n z, q, χ. 2.3 Proof. We use the method in 26, 27 for the proof. First, consider the case χ n  1, which implies χ  w n .Then b n  − z, q −1 ,χ   β n,q −1 ,1  − p ∗ z  − p n−1 q −1 β n,q −p ,1  − p −1 p ∗ z   β n,q −1  1 − p ∗ z  − 1  q p−1  n−1 p n−1 q β n,q −p  1 − p −1 p ∗ z  . 2.4 From 1.23,wehave b n  − z, q −1 ,χ  −1 n q n−1 β n,q  p ∗ z  − p n−1 q  q p−1  n−1 −1 n  q p  n−1 β n,q p  p −1 p ∗ z  −1 n q n−1  β n,q  p ∗ z  − p n−1 q β n,q p  p −1 p ∗ z   . 2.5 Using 1.24,weobtain b n  − z, q −1 ,χ  −1 n q n−1  β n,q  1p ∗ z  −nq p ∗ z  p ∗ z  n−1 q −p n−1 q β n,q p  1p −1 p ∗ z  p n−1 q n  q p  p −1 p ∗ z  p −1 p ∗ z  n−1 q p  −1 n q n−1  β n,q  1  p ∗ z  − p n−1 q β n,q p  1  p −1 p ∗ z   −1 n q n−1  β n,q,1  p ∗ z  − p n−1 q β n,q p ,1  p −1 p ∗ z   −1 n q n−1 b n z, q, χ. 2.6 Since χ  w n and w−11, the lemma holds for χ n  1. M. Cenkci and V. Kurt 9 Now, suppose that χ n /  1. Then, from 1.26,weobtain b n  − z, q −1 ,χ   β n,q −1 ,χ n  − p ∗ z  − χ n pp n−1 q −1 β n,q −p ,χ n  − p −1 p ∗ z    f χ n  n−1 q −1 f χ n  a1 χ n aβ n,q −f χ n  a − p ∗ z f χ n  − χ n pp n−1 q −1  f χ n  n−1 q −p f χ n  a1 χ n aβ n,q −pf χ n  a − p −1 p ∗ z f χ n    f χ n  n−1 q −1 f χ n  a1 χ n  f χ n − a  β n,q −f χ n  f χ n − a − p ∗ z f χ n  − χ n pp n−1 q −1  f χ n  n−1 q −p f χ n  a1 χ n  f χ n − a  β n,q −pf χ n  f χ n − a − p −1 p ∗ z f χ n    f χ n  n−1 q −1 f χ n  a1 χ n −aβ n,q −f χ n  1 − a  p ∗ z f χ n  − χ n pp n−1 q −1  f χ n  n−1 q −p f χ n  a1 χ n −aβ n,q −pf χ n  1 − a  p −1 p ∗ z f χ n  . 2.7 Using 1.23,wehave b n  − z, q −1 ,χ  −1 n  q f χ n  n−1  f χ n  n−1 q −1 χ n −1 f χ n  a1 χ n aβ n,q f χ n  a  p ∗ z f χ n  − χ n pp n−1 q −1 −1 n  q f χ n  n−1  f χ n  n−1 q −p χ n −1 f χ n  a1 χ n aβ n,q pf χ n  a  p −1 p ∗ z f χ n  −1 n q n−1 χ n −1β n,q,χ n  p ∗ z  − χ n pp n−1 q −1 n q n−1 χ n −1β n,q p ,χ n  p −1 p ∗ z  −1 n q n−1 χ n −1b n z, q, χ. 2.8 Note that χ n −1−1 n χ−1. Thus, the lemma holds for χ n /  1. Since the lemma holds for χ n  1andχ n /  1, the proof must be complete. Using this result, we can prove the following theorem. Theorem 2.2. Let z ∈ C p , |z| p ≤ 1,ands ∈ D, except for s /  1 if χ  1.Then L p,q −1 s, −z, χχ−1q −s L p,q s, z, χ. 2.9 Proof. Let z ∈ C p , |z| p ≤ 1, and n ∈ Z, n ≥ 1. Since L p,q 1 − n, z, χ− 1 n b n z, q, χ. 2.10 10 Journal of Inequalities and Applications Lemma 2.1 implies that L p,q −1 1 − n, −z, χ  − 1 n b n  − z, q −1 ,χ   − 1 n χ−1q n−1 b n z, q, χχ−1q n−1 L p,q 1 − n, z, χ, 2.11 and 2.9 holds for all s  1 − n, n ∈ Z, n ≥ 1. Since the negative integers have 0 as a limit point, Lemma 1.7 implies that Theorem 2.2 holds for all s in any neighborhood about 0 common to the domains of the functions on either side of 2.9. It is obvious that the domains, in the variable s, of the functions on both sides of 2.9 contain D,exceptfors /  1ifχ  1. This completes the proof. It is well known that the generalized Bernoulli polynomials associated with a Dirichlet character χ are important in regard to sums of consecutive integers, all of which raised to the same power. Proposition 1.6 represents a q-extension of this property. In this section, we will give an extension of Proposition 1.6 with the use of L p,q s, z, χ. For the character χ,letF 0  lcmf, p ∗ . Then, f χ n | F 0 for each n ∈ Z. Also, let F be a positive multiple of pp ∗  −1 F 0 . Theorem 2.3. Let z ∈ C p , |z| p ≤ 1,ands ∈ D, except for s /  1 if χ  1.Then L p,q s, z  F, χ − L p,q s, z, χ− p ∗ F  a1 a,p1 χ 1 aq ap ∗ z  a  p ∗ z : q  −s . 2.12 Proof. Let z ∈ C p , |z| p ≤ 1, and let n ∈ Z, n ≥ 1. From 2.1,wehave L p,q 1 − n, z  F, χ − L p,q 1 − n, z, χ− 1 n  b n z  F, q, χ − b n  z, q, χ. 2.13 Equation 2.2 then implies that b n z  F, q, χ − b n z, q, χ   β n,q,χ n  p ∗ zp ∗ F  − β n,q,χ n  p ∗ z   − χ n pp n−1 q  β n,q p ,χ n  p −1 p ∗ zp −1 p ∗ F  − β n,q p ,χ n  p −1 p ∗ z   . 2.14 By Proposition 1.6,wecanwrite b n z  F, q, χ − b n z, q, χ  n p ∗ F  a1 χ n aq ap ∗ z  a  p ∗ z  n−1 q − χ n pp n−1 q n p −1 p ∗ F  a1 χ n a  q p  ap −1 p ∗ z  a  p −1 p ∗ z  n−1 q p  n p ∗ F  a1 χ n aq ap ∗ z  a  p ∗ z  n−1 q − n p ∗ F  a1 p|a χ n aq ap ∗ z  a  p ∗ z  n−1 q  n p ∗ F  a1 a,p1 χ n aq ap ∗ z  a  p ∗ z  n−1 q . 2.15 [...]... of 2.12 have domains which contain D, except possibly for s / 1 if χ 1 This completes the proof Corollary 2.4 For s ∈ D, except for s / 1 if χ 1 Then p∗ F Lp,q s, F, χ − Lp,q s, χ χ1 a q a a : q − a 1 a,p 1 −s 2.23 12 Journal of Inequalities and Applications 3 Congruences for generalized q-Bernoulli polynomials Congruences related to classical and generalized Bernoulli numbers have found an amount of... it has been shown that for similar character χ, p−1 Δc k Bn,χn ∈ Zp , n 1 − χn p pn−1 3.10 and this value, modulo pZp , is independent of n Fox 6 derived congruences similar to those above for the generalized Bernoulli polynomials without restrictions on the character χ We now consider how Corollary 2.4 can be utilized to derive a collection of congruences related to generalized q-Bernoulli polynomials... generating function log 1 t k! k ∞ tn , n! s n, k n 0 3.31 0 for k ∈ Z, k ≥ 0 Since there is no constant term in the expansion of log 1 t , s n, k for 0 ≤ n < k Also, s n, n 1, for all n ≥ 0 The numbers s n, k are integers and satisfy the following relation related to binomial coefficients: 1 n s n, k xk n! k 0 x k 3.32 For further information for Stirling numbers, we refer to 32 −1 Theorem 3.3 Let n,... but with n > k Kummer congruences for generalized Bernoulli numbers Bn,χ were first regarded by Carlitz 28 For positive c ∈ Z, c ≡ 0 mod p − 1 , n, k ∈ Z, n > k ≥ 1, and χ such that f fχ / pm , where m ∈ Z, m ≥ 0, p−k Δk c Bn,χ ∈ Zp χ n 3.5 Here, Zp χ denotes the ring of polynomials in χ, whose coefficients are in Zp Shiratani 29 applied the operator Δk to − 1 − χn p pn−1 Bn,χn /n for similar c and χ,... We incorporate the polynomial structure Bn z, q, χ − 1 βn,q,χn p∗ z − χn p p n n−1 p q βn,q ,χn p−1 p∗ z 3.11 and the set structure R∗ x ∈ Zp : |x|p < p−1/ p−1 3.12 to derive the Kummer congruences for generalized q-Bernoulli polynomials Throughout, we assume that q ∈ Zp with |1 − q|p < p−1/ p−1 , so that q ≡ 1 mod R∗ −1 Theorem 3.1 Let n, c, k be positive integers and z ∈ p p∗ p∗ −k Δk Bn z, q, χ... examples is the Kummer congruences for classical Bernoulli numbers cf 2 : p−1 Δc Bn ∈ Zp , n 3.1 where c ∈ Z, c ≥ 1, c ≡ 0 mod p − 1 , and n ∈ Z is positive, even, and n / 0 mod p − 1 ≡ Here, Δc is the forward difference operator which operates on a sequence {xn } by Δc xn The powers Δk of Δc are defined by Δ0 c c so that Δk xn c xn c − xn 3.2 identity and Δk c k m 0 k m −1 k−m xn Δc ◦ Δk−1 for positive integers... p-adic L-functions at s 0,” Inventiones Mathematicae, vol 50, no 1, pp 91–102, 1978 16 L Carlitz, q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol 15, no 4, pp 987– 1000, 1948 17 M Cenkci, M Can, and V Kurt, “p-adic interpolation functions and Kummer-type congruences for q-twisted and q -generalized twisted Euler numbers,” Advanced Studies in Contemporary Mathematics, vol 9, no 2, pp... Integral Transforms and Special Functions, vol 15, no 5, pp 415–420, 2004 27 T Kim and S.-H Rim, “A note on the q-integral and q-series,” Advanced Studies in Contemporary Mathematics, vol 2, pp 37–45, 2000 28 L Carlitz, “Arithmetic properties of generalized Bernoulli numbers,” Journal fur die Reine und Ange¨ wandte Mathematik, vol 202, pp 174–182, 1959 29 K Shiratani, “Kummer’s congruence for generalized. .. ∈ R∗ χ c 3.22 Furthermore, for a positive integer n , lim j→∞ p∗ p∗ −k −k Δk Bn zj , q, χ − p∗ c Δk Bn z, q, χ − p∗ c −k −k Δk Bn 0, q, χ − c Δk Bn 0, q, χ c − p∗ −k p∗ −k Δk Bn zj , q, χ − p∗ c Δk Bn z, q, χ − p∗ c −k −k Δk Bn 0, q, χ c Δk Bn 0, q, χ c 3.23 M Cenkci and V Kurt 15 Since zj ∈ p p∗ −1 F0 Z for all j, the quantity on the left must be 0 modulo p∗ R∗ χ Therefore, the value p∗ −k Δk Bn... Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol 12, no 2, pp 241–268, 2005 25 T Kim and S.-H Rim, “A note on p-adic Carlitz’s q-Bernoulli numbers,” Bulletin of the Australian Mathematical Society, vol 62, no 2, pp 227–234, 2000 26 T Kim, “p-adic q-integrals associated with the Changhee-Barnes’ q-Bernoulli . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 270713, 19 pages doi:10.1155/2008/270713 Research Article Congruences for Generalized q-Bernoulli Polynomials Mehmet Cenkci and Veli. of these proper- ties yields general classes of congruences for generalized q-Bernoulli polynomials, which are q- extensions of the classes for generalized Bernoulli numbers and polynomials given. 2.23 12 Journal of Inequalities and Applications 3. Congruences for generalized q-Bernoulli polynomials Congruences related to classical and generalized Bernoulli numbers have found an amount of