Algorithms to solve coupled systems of differential equations in terms of power series

TL;DR

This paper develops algorithms for representing solutions of coupled differential equations, especially from Feynman integrals, as nested sums over hypergeometric products, enabling constructive power series solutions with initial values.

Contribution

It introduces new algorithms and variations for solving coupled differential systems by reducing them to scalar recurrences within nested sums, improving computational efficiency.

Findings

Successfully applied to 3-loop Feynman integrals

Reduced recurrence order in the new tactic

Demonstrated efficiency with existing summation technologies

Abstract

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their power series representations can be given within the class of nested sums over hypergeometric products. In this article we will work out the calculation steps that solve this problem. First, we will present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second, we will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lower recurrence orders. The article will work out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies.