Extensions of the AZ-algorithm and the Package MultiIntegrate

TL;DR

This paper extends the multivariate Almkvist-Zeilberger algorithm to handle complex Feynman integrals in quantum field theory, enabling the computation of closed forms or series expansions for multidimensional hyperexponential integrals.

Contribution

It introduces new methods and algorithms for the extended AZ-algorithm and presents an enhanced Mathematica package, MultiIntegrate, for computing recurrences and series expansions of hyperexponential integrals.

Findings

Successfully computes recurrences and differential equations for complex integrals.

Enables closed form or series expansion representations of multidimensional integrals.

Integrates with Sigma and HarmonicSums for comprehensive symbolic computation.

Abstract

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.