# Generalized Fourier Series for Solutions of Linear Differential Equations

**Generalized Fourier Series for Solutions of Linear Differential Equations**

Chebyshev polynomials, Hermite polynomials, Bessel functions and
other families of special functions each form a basis of some Hilbert
space. A Generalized Fourier Series is a series expansion in one of
these bases, for instance a Chebyshev series. 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 us
give a general algorithm for computing this recurrence, and a simple
view of existing algorithms for several specific function families.

Joint work with Bruno Salvy.