A note on the zeros and local extrema of Digamma related functions

TL;DR

This paper investigates the zeros and local extrema of the Digamma function and related functions, providing new representations, sums, and approximations, including the appearance of Lambert functions in hyperfactorial estimates.

Contribution

It introduces Weierstrassian product representations for the Digamma function and explores zeros of related functions, offering new insights and accurate approximations.

Findings

New product representations for the Digamma function

Identified interesting sums of zeros of the Digamma and Barnes G functions

Derived accurate hyperfactorial approximations involving Lambert functions

Abstract

Little is known about the zeros of the Digamma function. Establishing some Weierstrassian infinite product representations for a given regularization of the Digamma function we find interesting sums of its zeros. In addition, we study the same questions for the zeros of the logarithmic derivative of the Barnes $G$ function. At the end of the paper we provide rather accurate approximations of the hyperfactorial where, rather interestingly, the Lambert function appears.