The double gamma function and Vladimir Alekseevsky

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 $\sin \pi x$ and taking half of the factors corresponding to non-positive roots of the sine, we obtain the function $1/\Gamma(x)$. 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 $n_1\omega+n_2\omega_2$ 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).