Zeros of varying Laguerre–Krall orthogonal polynomials

Abstract

In this paper we introduce a sequence of varying orthogonal polynomials related to a Laguerre weight where this absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the origin. The main objective is to obtain asymptotic relations between the zeros of these polynomials and the zeros of the Bessel functions of the first kind (or linear combinations of them). This is done through Mehler-Heine type formulas. With these relations we can easily compute asymptotically the zeros of these polynomials. We show some numerical experiments.

A differential equation for varying Krall-type orthogonal polynomials

“…The particular case with c = 0 and m = 1 was introduced in [6] and more generally in [32] where the authors obtained asymptotic properties of orthogonal polynomials with respect to (40) and properties about their zeros. It is important to remark that the holonomic equation for the orthogonal polynomials with respect to (40) was obtained in [22] in the constant case.…”