On Digamma Series Convertible Into Hypergeometric Series

Abstract

Series containing the digamma function arise when calculating parametric derivatives of hypergeometric functions, and play a role in evaluation of Feynman diagrams. As these series are typically non-hypergeometric, a few instances when they are summable in terms of hypergeometric functions are of importance. In this paper, by employing appropriate limiting processes, we convert multi-term identities for the generalized hypergeometric functions evaluated at positive/negative unity into identities connecting them to digamma series. The resulting formulas can be viewed as hypergeometric expressions for the sum of the partial derivatives of the generalized hypergeometric function with respect to all its parameters, and seem to have no direct analogues in the literature.