Double integrals and transformation formulas for Appell--Lauricella hypergeometric functions $F_D$

TL;DR

This paper derives transformation formulas for Appell--Lauricella hypergeometric functions using monodromy group analysis and double integral techniques, offering new insights and alternative proofs for classical transformations.

Contribution

It introduces a novel approach to derive transformation formulas for hypergeometric functions through monodromy and double integrals, expanding the understanding of their properties.

Findings

Derived new transformation formulas for Appell--Lauricella functions.

Provided an alternative proof for Goursat's quadratic transformations.

Connected monodromy group properties with integral representations.

Abstract

The monodromy of hypergeometric functions can govern the properties of the functions themselves. Previously, the second and third authors studied the commensurability relations among monodromy groups of the Appell--Lauricella hypergeometric functions using Deligne--Mostow theory and the geometric correspondence between curves and surfaces. In this paper, we apply the same construction to obtain transformation formulas among these hypergeometric functions. This also provides an alternative approach to some of Goursat's quadratic transformations via double integrals and Fubini's theorem.