On Some Sums of Digamma and Polygamma functions - Version (2017) and Review

TL;DR

This paper reviews and extends formulas involving sums of digamma and polygamma functions, correcting errors and providing a comprehensive resource for researchers needing these functions in applications.

Contribution

It compiles, reviews, and develops new formulas for sums of digamma and polygamma functions, including corrections of existing literature errors.

Findings

Collected and listed formulas involving sums of digamma and polygamma functions.

Corrected several errors present in previous literature.

Provided developments enabling formal proofs for these formulas.

Abstract

This paper is an enhanced version of a more than decade-older paper with a similar title. Many formulae involving both finite and infinite sums of digamma and polygamma functions up to quadratic order, few of which appear in standard reference works or the literature, but which periodically arise in applications, are collected, reviewed, listed and developed to the point that a knowledgeable reader could devise a formal proof. Several errors in the literature are corrected