Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

Abstract

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.