Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions

TL;DR

This paper extends Hermite reduction to linear differential operators, enabling efficient computation of operators satisfied by definite integrals of D-finite functions, thus advancing symbolic integration techniques.

Contribution

It introduces a generalized Hermite reduction for arbitrary linear differential operators and applies it to creative telescoping for D-finite functions.

Findings

Efficient algorithms for generalized Hermite reduction.

New methods for computing differential operators for definite integrals.

Enhanced symbolic integration capabilities for D-finite functions.

Abstract

Hermite reduction is a classical algorithmic tool in symbolic integration. It is used to decompose a given rational function as a sum of a function with simple poles and the derivative of another rational function. We extend Hermite reduction to arbitrary linear differential operators instead of the pure derivative, and develop efficient algorithms for this reduction. We then apply the generalized Hermite reduction to the computation of linear operators satisfied by single definite integrals of D-finite functions of several continuous or discrete parameters. The resulting algorithm is a generalization of reduction-based methods for creative telescoping.