Chebyshev Expansions for Solutions of Linear Differential Equations

A Chebyshev expansion is a series in the basis of Chebyshev polynomials of the first kind. When such a series solves a linear differential equation, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets its give a simple view of previous algorithms, analyze their complexity, and design a faster one for large orders.