Báo cáo sinh học: Some results for the q-Bernoulli, q-Euler numbers and polynomials

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Báo cáo sinh học: " Some results for the q-Bernoulli, q-Euler numbers and polynomials"Advances in DifferenceEquations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Some results for the q-Bernoulli, q-Euler numbers and polynomials Advances in Difference Equations 2011, 2011:68 doi:10.1186/1687-1847-2011-68 Daeyeoul Kim (daeyeoul@nims.re.kr) Min-Soo Kim (minsookim@kaist.ac.kr) ISSN 1687-1847 Article type Research Submission date 2 September 2011 Acceptance date 23 December 2011 Publication date 23 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/68 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Kim and Kim ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some results for the q -Bernoulli, q -Euler numbers and polynomials Daeyeoul Kim1 and Min-Soo Kim∗2 1 National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu Daejeon 305-340, South Korea 2 Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, South Korea ∗ Corresponding author: minsookim@kaist.ac.kr Email address: DK: daeyeoul@nims.re.kr Abstract The q -analogues of many well known formulas are derived by us- ing several results of q -Bernoulli, q -Euler numbers and polynomials. The q - analogues of ζ -type functions are given by using generating functions of q - Bernoulli, q -Euler numbers and polynomials. Finally, their values at non- positive integers are also been computed. 2010 Mathematics Subject Classification: 11B68; 11S40; 11S80. Keywords: Bosonic p-adic integrals; Fermionic p-adic integrals; q -Bernoulli polynomials; q -Euler polynomials; generating functions; q -analogues of ζ -type functions; q -analogues of the Dirichlet’s L-functions. 1. Introduction Carlitz [1,2] introduced q -analogues of the Bernoulli numbers and polynomials.From that time on these and other related subjects have been studied by variousauthors (see, e.g., [3–10]). Many recent studies on q -analogue of the Bernoulli,Euler numbers, and polynomials can be found in Choi et al. [11], Kamano [3], Kim[5,6,12], Luo [7], Satoh [9], Simsek [13,14] and Tsumura [10]. For a fixed prime p, Zp , Qp , and Cp denote the ring of p-adic integers, the fieldof p-adic numbers, and the completion of the algebraic closure of Qp , respectively.Let | · |p be the p-adic norm on Q with |p|p = p−1 . For convenience, | · |p will alsobe used to denote the extended valuation on Cp . The Bernoulli polynomials, denoted by Bn (x), are defined as n n Bk xn−k ,(1.1) Bn (x) = n ≥ 0, k k=0where Bk are the Bernoulli numbers given by the coefficients in the power series ∞ tk t(1.2) = Bk . et − 1 k! k=0 ...