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arXiv · 2604.26944

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

Alexandre Benoit·Nicolas Brisebarre·Bruno Salvy
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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.

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Alexandre Benoit, Nicolas Brisebarre, Bruno Salvy. 2026-04-29. Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials. https://arxiv.org/abs/2604.26944

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