On the Structure of Compatible Rational Functions

TL;DR

This paper characterizes the structure of compatible rational functions satisfying certain functional equations, providing a decomposition theorem, an algorithm, and an application in symbolic computation.

Contribution

It introduces a theorem that describes the structure of compatible rational functions and an algorithm to compute their decomposition.

Findings

Decomposition of solutions into rational, symbolic powers, hyperexponential, hypergeometric, and q-hypergeometric functions.

An algorithm for computing the decomposition.

Application demonstrating the method's utility.

Abstract

A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the structure of compatible rational functions. The theorem enables us to decompose a solution of such a system as a product of a rational function, several symbolic powers, a hyperexponential function, a hypergeometric term, and a q-hypergeometric term. We outline an algorithm for computing this product, and present an application.