On digamma series convertible into hypergeometric series

Series containing the digamma function arise when calculating the parametric derivatives of the 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, we convert multi-term identities for the generalized hypergeometric functions evaluated at unity into identities connecting them to the digamma series via the appropriate limiting process. The resulting formulas can be viewed as hypergeometric expressions for the $1$-norm of the gradient of the generalized hypergeometric function with respect to all its parameters and seem to have no direct analogues in the literature.