On one dimensional digamma and polygamma series related to the evaluation of Feynman diagrams

Abstract

We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^nψ(n+p/q)/n^r and (\pm 1)^nψ^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including evaluation of Feynman amplitudes.