Algebraicity of the Appell-Lauricella and Horn hypergeometric functions

TL;DR

This paper extends classical results on algebraic Gauss functions to multivariable hypergeometric functions, showing that certain families like Lauricella F_C have infinitely many algebraic instances across any number of variables.

Contribution

It generalizes Schwarz's list to include Appell-Lauricella and Horn functions, demonstrating the existence of infinitely many algebraic functions in these classes.

Findings

Extension of Schwarz's list to multivariable functions

Identification of infinitely many algebraic Lauricella F_C functions

Comprehensive classification of algebraic Appell-Lauricella and Horn functions

Abstract

We extend Schwarz' list of irreducible algebraic Gauss functions to the four classes of Appell-Lauricella functions in several variables and the 14 complete Horn functions in two variables. This gives an example of a family of functions such that for any number of variables there are infinitely many algebraic functions, namely the Lauricella $F_C$ functions.