Zero distribution of orthogonal polynomials on a $q$-lattice

TL;DR

This paper analyzes the asymptotic distribution of zeros of orthogonal polynomials supported on a $q$-lattice, revealing their behavior through potential theory and equilibrium measures.

Contribution

It provides the first detailed asymptotic analysis of zeros of orthogonal polynomials on a $q$-lattice, connecting them to equilibrium measures in potential theory.

Findings

Zeros follow a specific asymptotic distribution related to equilibrium measures

Distribution is characterized by the radial part of an extremal problem

Results extend understanding of orthogonal polynomials on discrete supports

Abstract

We give the asymptotic behavior of the zeros of orthogonal polynomials, after appropriate scaling, for which the orthogonality measure is supported on the $q$-lattice $\{q^k, k=0,1,2,3,\ldots\}$, where $0 < q < 1$. The asymptotic distribution of the zeros is given by the radial part of the equilibrium measure of an extremal problem in logarithmic potential theory for circular symmetric measures with a constraint imposed by the $q$-lattice.