On asymptotic behaviour of some series dependent on parameter

Abstract

The question studied concerns the behaviour, as α→∞, of the functions defined by the series with general terms a_k=k^−ν(k+α)⁻¹ and a_k=k^−ν(k−α)⁻¹, k=1,2,… , where the parameter $\alpha\not=k$ and the value ν>0. The asymptotic expansions of such functions in terms of powers of α⁻¹ when α→∞ are found (a logarithmical factor is also present in the only term when ν is integer). It is shown that the coefficients of the asymptotic expansions obtained are determined by values of the Riemann zeta-function ζ(z) on the sequence of points ν−m, where m=0,1,… and m≠ν−1. The Mellin transform technique, the Cauchy theorem on residues, and the recurrence formulae, connecting the series in question for the values of parameter ν and ν+1, are employed when deriving the asymptotic expansions.