Distributions of zeros of discrete and continuous polynomials from their recurrence relation

摘要

The hypergeometric polynomials in a continuous or a discrete variable, whose canonical forms are the so-called classical orthogonal polynomial systems, are objects which naturally appear in a broad range of physical and mathematical fields from quantum mechanics, the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of Sturm–Liouville differential and difference equations. Often, they are encountered in the form of a three-term recurrence relation (TTRR) which connects a polynomial of a given order with the polynomial of the contiguous orders. This relation can be directly found, in particular, by use of Lanczos-type methods, tight-binding models or the application of the conventional discretization procedures to a given differential operator. Here the distribution of zeros and its asymptotic limit, characterized by means of its moments around the origin, is described for the continuous classical (Hermite, Laguerre, Jacobi, Bessel) polynomials and for the discrete classical (Charlier, Meixner, Kravchuk, Hahn) polynomials by means of 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. Let us underline that the orthogonality condition is not required in our approach. The moments are given in an explicit manner which, at times, allow us to recognize the analytical form of the corresponding distribution.