Schwarz's map for Appell's second hypergeometric system with quarter integer parameters

TL;DR

This paper investigates Schwarz's map for a specific Appell hypergeometric system with quarter-integer parameters, providing explicit equations and inverse mappings using theta functions, enhancing understanding of its geometric structure.

Contribution

It introduces a new explicit description of Schwarz's map for Appell's second system with specific parameters using theta functions, and derives its defining equations and inverse.

Findings

Explicit defining equations of the image set in complex space and upper half-plane.

Expression of the inverse Schwarz's map using theta functions.

Enhanced geometric understanding of the hypergeometric system's image.

Abstract

We study Schwarz's map for Appell's second system $\cF_2$ of hypergeometric differential equations in two variables with parameters $a=c_1=c_2=\frac{1}{2}$, $b_1=b_2=\frac{1}{4}$. By using theta functions with characteristics, we give a defining equation of an analytic set in $\C^2\times \H$ of its image, and express its inverse.