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Following the Mellin and inverse Mellin transform techniques presented in ourpaper arXiv:1606.02150 (NT), we have established close forms of Laurent seriesexpansions 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 whichinclude generalized harmonic numbers of first, second and third powers andvalues of the Riemann zeta-functions at integers / Bernoulli numbers, forexample 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.

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