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.