We present our investigation of the study of two variable hypergeometric series, namely Appell $F_{1}$ and $F_{3}$ 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 $F_{D}^{(3)}$ series and the Lauricella-Saran $F_{S}^{(3)}$ series, leveraging the analytic continuations of $F_{1}$ and $F_{3}$, 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: $\texttt{AppellF1.wl}$, $\texttt{AppellF3.wl}$, $\texttt{LauricellaFD.wl}$, and $\texttt{LauricellaSaranFS.wl}$ in $\textit{MATHEMATICA}$. 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 $\textit{MATHEMATICA}$ notebooks demonstrating different numerical results are also provided along with this paper.

翻译：我们系统研究了双变量超几何级数——即阿佩尔$F_{1}$和$F_{3}$级数，并获得了覆盖整个实$(x,y)$平面（除奇异轨迹外）的完整解析延拓列表。进一步地，基于$F_{1}$和$F_{3}$的解析延拓，我们推导了其三变量推广形式——劳里切拉$F_{D}^{(3)}$级数和劳里切拉-萨兰$F_{S}^{(3)}$级数的解析延拓，从而确保在除去函数奇异轨迹后，整个实$(x,y,z)$空间均可覆盖。这些研究源于多变量超几何函数在费曼积分评估中的频繁出现，同时也可应用于数学物理其他分支的相关场景。为便于实际应用，我们提供了四个$\textit{MATHEMATICA}$软件包：$\texttt{AppellF1.wl}$、$\texttt{AppellF3.wl}$、$\texttt{LauricellaFD.wl}$和$\texttt{LauricellaSaranFS.wl}$。这些软件包适用于参数的一般值和非一般值情形，并充分考虑了其在费曼积分评估中的实用价值。我们以费曼积分评估为例，具体展示了这些软件包的多种物理应用场景，并通过其他方法对计算结果进行了交叉验证。本文还随附了多个展示不同数值结果的$\textit{MATHEMATICA}$笔记本文件。