Separability Problems in Creative Telescoping

TL;DR

This paper explores the separability problem in creative telescoping, providing criteria for testing separability in various classes of special functions, which is crucial for simplifying multivariate functions into univariate equations.

Contribution

It introduces new criteria for testing separability in algebraic, differential, and difference equations across multiple classes of special functions.

Findings

Criteria for rational functions and hyperexponential functions.

Methods for hypergeometric terms and algebraic functions.

Insights into the role of separability in creative telescoping.

Abstract

For given multivariate functions specified by algebraic, differential or difference equations, the separability problem is to decide whether they satisfy linear differential or difference equations in one variable. In this paper, we will explain how separability problems arise naturally in creative telescoping and present some criteria for testing the separability for several classes of special functions, including rational functions, hyperexponential functions, hypergeometric terms, and algebraic functions.