Analytic continuations of the Horn $H_1$ and $H_5$ functions

TL;DR

This paper derives the analytic continuations of the Horn H1 and H5 functions using symbolic computation, emphasizing transformations to extend their regions of convergence, and provides a Mathematica package for practical use.

Contribution

First derivation of analytic continuations for Horn H1 and H5 functions using automated symbolic tools and transformations, with publicly available implementation.

Findings

Derived ACs for Horn H1 and H5 functions.

Extended regions of convergence using Pfaff-Euler transformations.

Provided a Mathematica package with all results.

Abstract

The analytic continuations (ACs) of the double variable Horn $H_1$ and $H_5$ functions have been derived for the first time using the automated symbolic $\textit{Mathematica}$ package $\texttt{Olsson.wl}$. The use of Pfaff-Euler transformations have been emphasised to derive AC to cover regions which are otherwise not possible. The corresponding region of convergence (ROC) is obtained using its companion package $\texttt{ROC2.wl}$. A $\textit{Mathematica}$ package $\texttt{HornH1H5.wl}$, containing all the derived ACs and the associated ROCs, along with a demonstration file of the same is made publicly available in https://github.com/souvik5151/Horn_H1_H5 .