MultiHypExp: A Mathematica package for expanding multivariate hypergeometric functions in terms of multiple polylogarithms

We present the Mathematica package MultiHypExp that allows for the expansion of multivariate hypergeometric functions (MHFs), especially those likely to appear as solutions of multi-loop, multi-scale Feynman integrals, in the dimensional regularization parameter. The series expansion of MHFs can be carried out around integer values of parameters to express the series coefficients in terms of multiple polylogarithms. The package uses a modified version of the algorithm prescribed in [1]. In the present work, we relate a given MHF to a Taylor series expandable MHF by a differential operator. The Taylor expansion of the latter MHF is found by first finding the associated partial differential equations (PDEs) from its series representation. We then bring the PDEs to the Pfaffian system and further to the canonical form, and solve them order by order in the expansion parameter using appropriate boundary conditions. The Taylor expansion so obtained and the differential operators are used to find the series expansion of the given MHF. We provide examples to demonstrate the algorithm and to describe the usage of the package, which can be found in this url https://github.com/souvik5151/MultiHypExp.

本报告推出名为 MultiHypExp 的 Mathematica 软件包，该软件包允许多变量超几何函数（MHFs）在维度正则化参数中进行展开，尤其是那些可能作为多圈、多尺度费曼积分解出现的函数。MHFs 的级数展开可在参数的整数值附近进行，从而将级数系数表达为多个多对数函数。该软件包采用了文献[1]中规定的算法的改进版本。在本研究中，我们通过微分算子将给定的 MHF 与可泰勒级数展开的 MHF 相关联。通过首先从其级数表示中找出相关的偏微分方程（PDEs），进而发现后者的泰勒展开。然后，我们将 PDEs 转换为 Pfaffian 系统，并进一步转化为规范形式，在展开参数的各阶使用适当的边界条件逐阶求解。所获得的泰勒展开和微分算子被用来寻找给定 MHF 的级数展开。此外，我们提供了示例以展示算法，并描述了软件包的使用方法，相关信息可在以下网址找到：https://github.com/souvik5151/MultiHypExp。