On the Barnes double gamma function

TL;DR

This paper comprehensively reviews the Barnes double gamma function, introduces a numerical computation algorithm, and presents new properties and identities related to the function and associated modular forms.

Contribution

It consolidates known properties, develops a new numerical algorithm, and derives novel identities and representations for the Barnes double gamma function.

Findings

Complete asymptotic expansion as z→∞

New product identity for the double gamma function

New representations of gamma modular forms C(τ) and D(τ)

Abstract

We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function $G(z;\tau)$. Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as $z\to \infty$. Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms $C(\tau)$ and $D(\tau)$.