Carlitz’s q

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• 作者：Junesang Choi ; P.J. Anderson ; H.M. Srivastava
• 关键词：q-Extensions of the Riemann zeta function and the Hurwitz zeta function ; q-Extensions of the Bernoulli and Euler polynomials and numbers ; q-Stirling numbers of the second kind ; Euler– ; Maclaurin summation formula
• 刊名：Applied Mathematics and Computation
• 出版年：2009
• 出版时间：1 October 2009
• 年：2009
• 卷：215
• 期：3
• 页码：1185-1208
• 全文大小：387 K

文摘

In this paper, we systematically recover the identities for the q-eta numbers η_k and the q-eta polynomials η_k(x), presented by Carlitz [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000], which we define here via generating series rather than via the difference equations of Carlitz. Following a method developed by Kaneko et al. [M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler’s approach to the Riemann zeta function, Kyushu J. Math. 57 (2003) 175–192] for a canonical q-extension of the Riemann zeta function, we investigate a similarly constructed q-extension of the Hurwitz zeta function. The details of this investigation disclose some interesting connections among q-eta polynomials, Carlitz’s q-Bernoulli polynomials -polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here.