On a family of symmetric hypergeometric functions of several variables and their Euler type integral representation

TL;DR

This paper introduces a family of symmetric hypergeometric functions of multiple variables, representing them via multidimensional Euler integrals and exploring their algebraic properties and connections to classical special functions.

Contribution

It provides explicit integral representations of these hypergeometric functions and links them to existing theories like A-hypergeometric systems and elliptic functions.

Findings

Each function $G_n$ can be expressed through an integral involving $G_{n-1}$.

Functions $G_2$ and $G_3$ are algebraic or relate to elliptic functions.

The functions exhibit quasi-invariance under certain involutions.

Abstract

This paper is devoted to the family $\{G_n\}$ of hypergeometric series of any finite number of variables, the coefficients being the square of the multinomial coefficients $(\ell_1+...+\ell_n)!/(\ell_1!...\ell_n!)$, where $n\in\ZZ_{\ge 1}$. All these series belong to the family of the general Appell-Lauricella's series. It is shown that each function $G_n$ can be expressed by an integral involving the previous one, $G_{n-1}$. Thus this family can be represented by a multidimensional Euler type integral, what suggests some explicit link with the Gelfand-Kapranov-Zelevinsky's theory of $A$-hypergeometric systems or with the Aomoto's theory of hypermeotric functions. The quasi-invariance of each function $G_n$ with regard to the action of a finite number of involutions of $\CC^{*n}$ is also established. Finally, a particular attention is reserved to the study of the functions $G_2$ and $G_3$, each of which is proved to be algebraic or to be expressed by the Legendre's elliptic function of the first kind.