Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials

TL;DR

This paper introduces a unified approach to compute recurrence relations for series expansions in classical orthogonal polynomial bases, using fractions of recurrence operators and a noncommutative Euclidean algorithm.

Contribution

It provides a simple, unified framework for deriving recurrence relations and demonstrates its effectiveness through various examples.

Findings

Unified view of algorithms for recurrence computation

Effective in multiple classical orthogonal polynomial cases

Utilizes noncommutative Euclidean algorithm as core method

Abstract

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.