Olsson.wl : a Mathematica package for the computation of linear transformations of multivariable hypergeometric functions

TL;DR

The paper introduces Olsson.wl, a Mathematica package that automates the derivation of linear transformations and analytic continuations for multivariable hypergeometric functions by leveraging Olsson's method, which builds higher-variable continuations from transformations of $_2F_1$. It also presents ROC2.wl, a companion tool that automatically determines regions of convergence for double hypergeometric series using Horn-type theorems, enabling robust numerical evaluation. The authors demonstrate new analytic continuations for functions such as $F_2$, $F_4$, and Horn series $H_1$ and $H_5$, and validate these against known results and numerical checks, with extensions to Lauricella cases discussed. Collectively, these tools automate a broad class of transformations and convergence analyses, providing a practical framework for analytic and numerical work on multivariable hypergeometric functions in physics and mathematics, with plans to extend to higher-variable cases and to address multivaluedness and branch-cut complexities.

Abstract

We present the Olsson$.$wl Mathematica package which aims to find linear transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson in J. Math. Phys. 5, 420 (1964) in order to derive the analytic continuations of the Appell $F_1$ double hypergeometric series from the linear transformations of the Gauss $_2F_1$ hypergeometric function. We provide a brief description of Olsson's method and demonstrate the commands of the package, along with examples. We also provide a companion package, called ROC2$.$wl and dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson$.$wl.

Figures (1)

• Figure 1: Flow chart of the ROC2 algorithm

Theorems & Definitions (2)

• Theorem 1
• Theorem 2