Affiliations: Institute of Hydromechanics, National Academy of Sciences, Kiev, Ukraine E-mail: alex@gomilko.com | Kiev National Taras Shevchenko University, Kiev, Ukraine E-mail: ulitko@univ.kiev.ua
Note: [] Corresponding author.
Abstract: The question studied concerns the behaviour, as α→∞, of the functions defined by the series with general terms ak=k−ν(k+α)−1 and ak=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.
Keywords: series dependent on parameter, asymptotic expansion, Mellin transform, Riemann zeta-function
