The double gamma function and Vladimir Alekseevsky

TL;DR

The paper surveys the origin and history of the Barnes double gamma function, focusing on Vladimir Alekseevsky's 1888–89 work that introduced $G(x;\alpha)$ via the relations $G(x+1;\alpha)=\Gamma\left(\frac{x}{\alpha}\right)G(x;\alpha)$ and $G(x+\alpha;\alpha)=(2\pi)^{\frac{\alpha-1}{2}}\alpha^{-{(2x-1)}/{2}}\Gamma(x)G(x;\alpha)$ with $G(0;\alpha)=1$, and its subsequent development through Weierstrass-type products, integral representations, and the multiplication formula. It traces the historical arc from Kinkelin to Barnes and Shintani, noting a long hiatus before resurgence, and discusses Alekseevsky's biography and historiography to illustrate how significant ideas can be forgotten and later revived, with implications for modern understanding of special functions related to theta-functions and $q$-analogues.

Abstract

This paper is about a forgotten function and a forgotten mathematician. The double gamma function is now an important special function, which appears for different reasons in many branches of mathematics and in mathematical physics, as it satisfies a large and amusing collection of identities parallel to the classical gamma function. It was discovered and investigated in detail by Vladimir Alekseevsky in 1888-89. We outline his starting point here: considering the Weierstrass product for the entire periodic function $\sin πx$ and taking half of the factors corresponding to non-positive roots of the sine, we obtain the function $1/Γ(x)$. Considering the Weierstrass product for the doubly quasiperiodic Jacobi theta function $\vartheta_1$ and taking the half of factors we come to the $q$-gamma function (which appears, with a slight change in notation, in Eduard Heine's work). Considering the quarter of the factors (corresponding to the positive quadrant in the lattice $n_1ω+n_2ω_2$ of periods), Alekseevsky arrived at his double gamma function. The work in this direction was continued by Ernest Barnes, Jean Beaupin, Godfrey Hardy, and Vladimir Steklov in 1899--1907. After 1907, publications on this topic stopped, and the double gamma appeared again 70 years later due to Takuro Shintani. In this paper we discuss Alekseevsky's seminal work and its genesis, the history of the double gamma, and Alekseevsky's biography (1858-1916).