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

Sequences {pn}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 pnpn form sequences which converge to the eigenvalues of the corresponding differential equations. Moreover, interlacing properties of the zeros of pnpn are found. Applications to the numerical treatment of eigenvalue problems are given.