A connection between power series and Dirichlet series

Abstract

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.