An algebraic approach to certain differential eigenvalue problems (Q1915615)

The aim of the author is to derive a number of properties of the zeros of some orthogonal polynomials by considering only the recurrence relations satisfied by these polynomials. To this end he transforms first the known three-step recurrence relation connected with the given polynomial to a new recurrence relation which generates the characteristic polynomial of a symmetric tridiagonal matrix. Then he uses the already known properties of the eigenvalues of symmetric matrices. The method is explained on (and applied to) some particular polynomials such as Legendre, Krall-Legendre and Laguerre polynomials. For these polynomials he finds the intervals including the zeros, bounds for the zeros and approximations to the largest zeros. The paper includes also an example where the tridiagonal matrix is unsymmetric.