Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials

Sequences { p n } n = 0 ∞ of polynomials, orthogonal with respect to signed measures, are associated with a class of differential equations including the Mathieu, Lamé and Whittaker–Hill equation. It is shown that the zeros of p n form sequences which converge to the eigenvalues of the corresponding differential equations. Moreover, interlacing properties of the zeros of p n are found. Applications to the numerical treatment of eigenvalue problems are given.