Research Article On the Modified q-bernoulli Numbers of Higher Order with Weight

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1 Abstract and Applied Analysis Volume 202, Article ID , 6 pages doi:0.55/202/ Research Article On the Modified -Bernoulli Numbers of Higher Order with Weight T. Kim, J. Choi, 2 Y.-H. Kim, 2 and S.-H. Rim 3 Department of Mathematics, Kwangwoon University, Seoul 39-70, Republic of Korea 2 Division of General Education, Kwangwoon University, Seoul 39-70, Republic of Korea 3 Department of Mathematics Education, Kyungpook National University, Daegu , Republic of Korea Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr Received August 20; Revised 24 November 20; Accepted 3 December 20 Academic Editor: Ferhan M. Atici Copyright 202 T. 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. The purpose of this paper is to give some properties of the modified -Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic p-adic -integral on, we derive new identities of -Bernoulli numbers and polynomials with weight.. Introduction Let p be a fixed odd prime number. Throughout this paper, Q p,andc p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of Q p.letnbe the set of natural numbers and Z N {0}. Thep-adic norm of C p is defined by p p /p. When one talks of a -extension, can be considered as an indeterminate, a complex number C, orap-adic number C p. Throughout this paper we assume that α Q and C p with p <p /p so that x expx log. Let UD be the space of uniformly differentiable functions on. For f UD, the p-adic -integral on is defined by Kim see 3 as follows: I f ) fxdμ x lim N p [ ] N fx x, p N x0. where x is the -number of x which is defined by x x /.

2 2 Abstract and Applied Analysis From., we have n ) ) ) n I fn I f l fl n l f l, log.2 where f n x fx n see 2 4. As is well known, Bernoulli numbers are inductively defined by B 0, { if n, B n B n 0 if n>,.3 with the usual convention about replacing B n by B n see 3, 5. In 2, 5, 6,the-Bernoulli numbers are defined by B 0, log, B ) { n if n, Bn, 0 if n>,.4 with the usual convention about replacing B n by B n,. Note that lim B n, B n.inthe viewpoint of.4, we consider the modified -Bernoulli numbers with weight. In this paper we study families of the modified -Bernoulli numbers and polynomials of higher order with weight. In particular, by using the multivariate p-adic -integral on, we give new identities of the higher-order -Bernoulli numbers and polynomials with weight. 2. Modified -Bernoulli Numbers with Weight of Higher Order For n Z, let us consider the following modified -Bernoulli numbers with weight α see, 3: n, n,x ) nα n x n α x dμ x [ ] n x y y ) dμ α y ) nα n ) n l αl, l αl ) n l αlx αl. l αl 2. From 2., wenotethat n,x ) n x n l l α αlx l, 2.2 see, 3.

3 Abstract and Applied Analysis 3 For k N and n Z, by making use of the multivariate p-adic -integral on,we consider the following modified -Bernoulli numbers with weight α of order k, n, : Note that B,α n, n, x x k n α x x k dμ x dμ x k. n, and lim n, B k n, where B k n are the nth ordinary Bernoulli numbers of order k. For k, N N, we have 2.3 ) k p N pn i 0 pn ) nα n p N i k 0 ) k α i i k n α ) n p N i,...,i k 0 ) n j αi i k j j n ) j αpn j j ) p N k ) αpnj. }{{} k times 2.4 By., 2.3, and2.4, weget n, ) nα n n ) j. 2.5 Therefore, by 2.5, we obtain the following theorem. Theorem 2.. For n 0, one has n, ) nα n n ) j. 2.6 Let us consider the modified -Bernoulli and polynomials with weight α of order k as follows: n, x x x x k n α x x k dμ x dμ x k. 2.7 By the same method of 2.5, we obtain the following theorem. Theorem 2.2. For n Z, one has n, x ) nα n n ) j αxj. 2.8

4 4 Abstract and Applied Analysis By Theorem 2.2, weget k x n, ) α n n αn α ) n n αn α ) n n ) j k x ) n j ) ) k j ) k x n ) j x k n αn k n, x. 2.9 Therefore, by 2.9, we obtain the following theorem. Theorem 2.3. For n Z, one has n, k x n αn k n, x, n, k n αn k n,. 2.0 From Theorem 2.3, wenotethat lim k x B k n, n k x, lim k n B k n, n. 2. Thus, we have B k n k n B k n, where B k n are the nth Bernoulli numbers of order k. From 2.3 and 2.7, we can derive the following euations: B l,α x lim k, N mk α m l mk α m l i,...,i l 0 m m l [ ] p N l m m i,...,i l 0 p N m i,...,i l 0 n,...,n l 0 x i i l mn n l k α [ ] x i i k l x x l m αm mx mx l dμ mx dμ mx k ) B l,α x i i l. k, m m 2.2 Therefore, by 2.2, we obtain the following theorem. Theorem 2.4. For k Z and l, m N, one has B l,α k, x mk α m l m i,...,i l 0 ) B l,α x i i l. 2.3 k, m m

5 Abstract and Applied Analysis 5 In particular, B l,α k, mx mk α m l m i,...,i l 0 B l,α k, m x i i l m ). 2.4 From.2, we can derive the following integral: fx x dμ x fx x dμ x log f 0, fx 2 x dμ x f x x dμ x log f fx x dμ x f 0 f ). log 2.5 Continuing this process, we obtain fx n x dμ x fx x dμ x n f l. log 2.6 By 2.6, weget x n m α x dμ x x m α x dμ x mα n α Therefore, by 2. and 2.7, we obtain the following theorem. Theorem 2.5. For n N and m Z, one has l m α αl. 2.7 B m,n α m, m α n l m α ααl. 2.8 In an analogues manner as the previous investigation 7 0, we can define a further generalization of modified -Bernoulli numbers with weight. Let χ be the Dirichlet character with conductor d N. Then the generalized -Bernoulli numbers with weight attached to χ can be defined as follows: n,χ, χxx n α x dμ x X dn α d d a0 χa a n, d d ). 2.9 We expect to investigate these objects in future papers. This definition n, was also given in a previous paper see 9.

6 6 Abstract and Applied Analysis Acknowledgments The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 20. References S. Araci, D. Erdal, and J. J. Seo, A study on the fermionic p-adic -integral representation on associated with weighted -Bernstein and -Genocchi polynomials, Abstract and Applied Analysis, vol. 20, Article ID , 7 pages, T. Kim, -Volkenborn integration, Russian Mathematical Physics, vol. 9, no. 3, pp , T. Kim, On the weighted -Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics, vol. 2, no. 2, pp , C. S. Ryoo and Y. H. Kim, A numerical investigation on the structure of the roots of the twisted -Euler polynomials, Advanced Studies in Contemporary Mathematics, vol. 9, no., pp. 3 4, L. Carlitz, -Bernoulli numbers and polynomials, Duke Mathematical Journal, vol. 5, pp , Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russian Mathematical Physics, vol. 7, no. 4, pp , T. Kim, Power series and asymptotic series associated with the -analog of the two-variable p-adic L-function, Russian Mathematical Physics, vol. 2, no. 2, pp , T. Kim, D. V. Dolgy, S. H. Lee, B. Lee, and S. H. Rim, A note on the modified -Bernoulli numbers and polynomials with weight α, Abstract and Applied Analysis, vol. 20, Article ID 54534, 8 pages, T. Kim, D. V. Dolgy, B. Lee, and S.-H. Rim, Identities on the Weighted -Bernoulli numbers of higher order, Discrete Dynamics in Nature and Society, vol. 20, Article ID 98364, 6 pages, H. M. Srivastava, T. Kim, and Y. Simsek, -Bernoulli numbers and polynomials associated with multiple -zeta functions and basic L-series, Russian Mathematical Physics, vol. 2, no. 2, pp , 2005.

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