Analytic continuations and numerical evaluation of the Appell F₁, F₃, Lauricella FD⁽³⁾ and Lauricella-Saran FS⁽³⁾ and their Application to Feynman Integrals

We present our investigation of the study of two variable hypergeometric series, namely Appell F₁ and F₃ series, and obtain a comprehensive list of its analytic continuations enough to cover the whole real $(x,y)$ plane, except on their singular loci. We also derive analytic continuations of their 3-variable generalization, the Lauricella FD⁽³⁾ series and the Lauricella-Saran FS⁽³⁾ series, leveraging the analytic continuations of F₁ and F₃, which ensures that the whole real $(x,y,z)$ space is covered, except on the singular loci of these functions. While these studies are motivated by the frequent occurrence of these multivariable hypergeometric functions in Feynman integral evaluation, they can also be used whenever they appear in other branches of mathematical physics. To facilitate their practical use, we provide four packages: AppellF1.wl, AppellF3.wl, LauricellaFD.wl, and LauricellaSaranFS.wl in MATHEMATICA. These packages are applicable for generic as well as non-generic values of parameters, keeping in mind their utilities in the evaluation of the Feynman integrals. We explicitly present various physical applications of these packages in the context of Feynman integral evaluation and compare the results using other means as well. Various MATHEMATICA notebooks demonstrating different numerical results are also provided along with this paper.