We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrands and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals.

中文翻译：

---

AZ算法的扩展和Package MultiIntegrate

我们扩展（连续）多元Almkvist-Zeilberger算法，以将其应用于例如可重归一化量子场理论中出现的特殊Feynman积分。我们将考虑超指数积分上的多维积分，并尝试根据嵌套的总和与乘积或迭代积分找到封闭形式的表示形式。另外，如果我们不能完全通用地计算封闭形式的解决方案，那么我们可能会成功地以不确定的嵌套和和乘积或迭代积分的形式来计算此类积分的Laurent级数展开的第一系数。在本文中，我们介绍了相应的方法和算法。我们的Mathematica软件包MultiIntegrate，可以被视为（连续）多元Almkvist Zeilberger算法的增强实现，该算法可以计算超指数积分和积分的递归或微分方程。连同求和包Sigma和包HarmonicSums一起，我们的包提供了一些方法来根据嵌套和或迭代积分来计算超指数积分上多维积分的闭合形式表示形式（或Laurent级数展开的系数）。