Periodic perturbations of unbounded Jacobi matrices II: Formulas for density

TL;DR

This paper derives formulas for the density of the orthogonality measure of orthonormal polynomials with unbounded recurrence coefficients, using limits of scaled Turán determinants or Christoffel functions, and provides asymptotic analysis and numerical examples.

Contribution

It introduces explicit formulas for the measure density involving limits of scaled Turán determinants and Christoffel functions for unbounded Jacobi matrices.

Findings

Derived formulas for measure density using Turán determinants and Christoffel functions.

Provided exact asymptotics of orthogonal polynomials.

Included numerical examples illustrating the formulas.

Abstract

We give formulas for the density of the measure of orthogonality for orthonormal polynomials with unbounded recurrence coefficients. The formulas involve limits of appropriately scaled Tur\'an determinants or Christoffel functions. Exact asymptotics of the polynomials and numerical examples are also provided.