Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials
For researchers in symbolic computation and special functions, this provides a unified algorithmic framework for a known problem, but the contribution is incremental.
The paper unifies previous algorithms for computing recurrence equations satisfied by coefficients of orthogonal polynomial series solutions to linear ODEs, using a fraction of recurrence operators and a noncommutative Euclidean algorithm. The approach is demonstrated on various examples.
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.