An algorithm of computing inhomogeneous differential equations for definite integrals

TL;DR

This paper presents an algorithm that computes inhomogeneous differential equations for definite integrals with parameters, utilizing D-module integration techniques and Gr"obner basis methods.

Contribution

It introduces a novel algorithm combining D-module theory and Gr"obner bases to efficiently derive differential equations for parameter-dependent integrals.

Findings

Algorithm successfully computes inhomogeneous differential equations for various integrals.

Utilizes Gr"obner basis method within D-module framework for improved computation.

Builds on Oaku's integration algorithm for D-modules.

Abstract

We give an algorithm to compute inhomogeneous differential equations for definite integrals with parameters. The algorithm is based on the integration algorithm for $D$-modules by Oaku. Main tool in the algorithm is the Gr\"obner basis method in the ring of differential operators.