Summation rules for the values of the Riemann zeta-function and generalized harmonic numbers obtained using Laurent developments of polygamma functions and their products (2112.04380v1)

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.