The Liouville–Neumann expansion in singular eigenvalue problems

We consider boundary value problems of the form { x y ″ + f ( x ) y ′ + [ g ( x ) + λ σ ( x ) ] y = 0 , x ∈ ( 0 , 1 ) , y ( 0 ) = α , α 1 y ( 1 ) + α 2 y ′ ( 1 ) = β , with f ′ , g and σ continuous in [ 0 , 1 ] , σ ( x ) ≠ 0 , α , β , α 1 , α 2 ∈ R and λ ∈ C . We use the Liouville–Neumann technique to design an algorithm that approximates the eigenvalues λ and eigenfunctions y ( x ) of the problem; that is, for every couple ( λ , y ( x ) ) of eigenvalues and eigenvectors of the problem, we give a sequence ( λ n , y n ( x ) ) that converges uniformly on x ∈ [ 0 , 1 ] to the solution ( λ , y ( x ) ) of that problem. In particular, when f ( x ) , g ( x ) and σ ( x ) are polynomials, y n ( 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.