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 for multivariable hypergeometric functions, based on Olsson's method from 1964, facilitating analytic continuation and convergence analysis.

Contribution

It provides the first implementation of Olsson's method in a computational tool, enabling systematic derivation of transformations for multivariable hypergeometric functions.

Findings

Successfully derived transformations for Appell $F_1$ functions.

Demonstrated the package's commands with practical examples.

Provided a companion package for convergence region analysis.

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.