The Liouville–Neumann expansion in singular eigenvalue problems

We consider boundary value problems of the form {xy″+f(x)y′+[g(x)+λσ(x)]y=0,x∈(0,1),y(0)=α,α1y(1)+α2y′(1)=β, with f′f′, gg and σσ continuous in [0,1],σ(x)≠0[0,1],σ(x)≠0, α,β,α1,α2∈Rα,β,α1,α2∈R and λ∈Cλ∈C. We use the Liouville–Neumann technique to design an algorithm that approximates the eigenvalues λλ and eigenfunctions y(x)y(x) of the problem; that is, for every couple (λ,y(x))(λ,y(x)) of eigenvalues and eigenvectors of the problem, we give a sequence (λn,yn(x))(λn,yn(x)) that converges uniformly on x∈[0,1]x∈[0,1] to the solution (λ,y(x))(λ,y(x)) of that problem. In particular, when f(x)f(x), g(x)g(x) and σ(x)σ(x) are polynomials, yn(x)yn(x) are also polynomials. This technique may also be used to approximate the zeros of solutions of regular singular second-order linear differential equations and, in particular, of special functions.