Inverses of gamma functions

TL;DR

This paper investigates the inverse of the gamma function, extending it to a Pick-function, and provides integral representations for gamma and related functions, answering a question by Uchiyama.

Contribution

It extends the inverse gamma function to a Pick-function and derives integral representations for gamma, double gamma, and Barnes G-functions.

Findings

Inverse gamma function extends to a Pick-function on increasing intervals.

Integral representations for gamma, double gamma, and Barnes G-functions are established.

Results apply to a class of entire functions of genus 2.

Abstract

Euler's Gamma function $\Gamma$ either increases or decreases on intervals between two consequtive critical points. The inverse of $\Gamma$ on intervals of increase is shown to have an extension to a Pick-function and similar results are given on the intervals of decrease, thereby answering a question by Uchiyama. The corresponding integral representations are described. Similar results are obtained for a class of entire functions of genus 2, and in particular integral representations for the double gamma function and the $G$-function of Barnes are found.