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

2026-04-29

AI Summary

This work addresses the efficient computation of linear recurrence relations satisfied by the generalized Fourier coefficients of functions that solve linear differential equations, expressed in bases of classical orthogonal polynomials. To this end, the authors propose a unified framework based on fractional representations of linear recurrence operators, wherein the desired recurrence relation is interpreted as the numerator of such a fraction. The core computational mechanism relies on a non-commutative Euclidean algorithm, which systematically integrates several existing methods into a coherent and streamlined approach. Both theoretical analysis and illustrative examples demonstrate that the proposed framework not only offers broad generality but also enhances computational efficiency and implementation simplicity compared to prior techniques.

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.

Problem

• recurrence operators
• generalized Fourier series
• classical orthogonal polynomials
• linear differential equations
• polynomial coefficients

Innovation

• recurrence operators
• orthogonal polynomials
• noncommutative Euclidean algorithm
• generalized Fourier series
• linear differential equations