Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials
2026-04-29 • Symbolic Computation
Symbolic Computation
AI summaryⓘ
The authors study series made from classical orthogonal polynomials that solve certain linear differential equations. They show that the coefficients in these series follow a pattern described by linear recurrence equations. By viewing these equations as fractions of linear recurrence operators, the authors provide a clearer way to understand and compute these patterns using a noncommutative version of the Euclidean algorithm. They also test their method on several examples to show how well it works.
orthogonal polynomialsseries expansionslinear differential equationspolynomial coefficientslinear recurrence equationsnoncommutative algebraEuclidean algorithmrecurrence operators
Authors
Alexandre Benoit, Nicolas Brisebarre, Bruno Salvy
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.