Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions

Abstract

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 β k ( x ) , ϵ -polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here.