Evaluation of terminating and non-terminating sums containing the digamma function

Abstract

We derive transformation and summation formulas for terminating and nonterminating series involving the digamma function. Our principal results are obtained by a limiting process starting with duality relations for the generalized hypergeometric functions and their consequences. Selected formulas are further extended by parameter differentiation of Euler's transformation and by using contiguous relations. Most of our identities express products of hypergeometric and digamma series in terms of hypergeometric functions, and some evaluations of terminating digamma sums involve Bernoulli polynomials. In several cases the digamma contributions cancel, producing identities involving only products of hypergeometric functions.