A connection between power series and Dirichlet series

Abstract

[EN] We prove that for any convergent Laurent series f(z) = ∞n=−k anzn with k ≥ 0, there is a meromorphic function F(s) on C whose only possible poles are among the integers n = 1, 2, ..., k, having residues Res(F; n) = a−n/(n − 1)!, and satisfying F(−n) = (−1)nn! an for n = 0, 1, 2, .... Under certain conditions, F(s) is a Mellin transform. In particular, this happens when f(z) is of the form H(e−z)e−z with H(z) analytic on the open unit disk. In this case, if H(z) = ∞ n=0 hnzn, the analytic continuation of H(z) to z = 1 is related to the analytic continuation of the Dirichlet series ∞n=1 hn−1n−s to the complex plane.