Distributions of zeros of discrete and continuous polynomials from their recurrence relation (Q1855669)

Here the distribution of zeros and its asymptotic limit, characterized by means of its moments around the origin is described for a general system of polynomials, defined by a three-term recurrence relation. This general system includes all classical orthogonal families of polynomials in the discrete (Hahn, Meixner, Kravchuk, Charlier) and continuous (Hermite, Laguerre, Jacobi, Bessel) cases. The authors have used a general procedure which (i) only requires the three-term recurrence relation and (ii) avoids the often high-brow subtleties of the potential theoretic considerations used in some recent approaches. The orthogonality condition is also not required in this approach. The moments are given in explicit manner, which, at times, allow us to recognize the analytical form of the corresponding distribution.