Summation rules for the values of the Riemann zeta-function and generalized harmonic numbers obtained using Laurent developments of polygamma functions and their products

TL;DR

This paper derives summation rules involving the Riemann zeta-function and generalized harmonic numbers using Laurent series expansions of polygamma functions, providing new closed-form relationships and numerical validations.

Contribution

It introduces new summation formulas connecting harmonic numbers and zeta-values via Laurent expansions of polygamma functions, expanding analytical tools in number theory.

Findings

Derived closed-form Laurent series expansions of polygamma function products

Established new summation rules involving harmonic numbers and zeta-values

Numerically validated some of the derived summation formulas

Abstract

Following the Mellin and inverse Mellin transform techniques presented in our paper arXiv:1606.02150 (NT), we have established close forms of Laurent series expansions of products of bi- and trigamma functions /psi(z)*/psi(-z) and /psi_(1)(z)*/psi_(1)(-z). These series were used to find summation rules which include generalized harmonic numbers of first, second and third powers and values of the Riemann zeta-functions at integers / Bernoulli numbers, for example 2*Sum_(k-1)^(infinity)(H_(k)^((2))/k^3)=6*/zeta(2)*/zeta(3)-9*/zeta(5). Some of these rules were tested numerically.