Finite and infinite hypergeometric sums involving the digamma function

TL;DR

This paper derives closed-form expressions for finite and infinite sums involving the digamma function using hypergeometric function differentiation techniques, providing new formulas and verifying them numerically.

Contribution

It introduces novel methods for evaluating sums with the digamma function via hypergeometric differentiation and compares different differentiation formulas for generalized hypergeometric functions.

Findings

Derived new closed-form sums involving the digamma function

Compared differentiation formulas for hypergeometric functions

Validated results through numerical checks

Abstract

We calculate some finite and infinite sums containing the digamma function in closed-form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative formulas of the Pochhammer symbol. Also, we compare two different differentiation formulas of the generalized hypergeometric function with respect to the parameters. For some particular cases, we recover some results found in the literature. Finally, all the results have been numerically checked.