Orthogonal polynomials with periodic recurrence coefficients

TL;DR

This paper investigates a special class of orthogonal polynomials with periodic recurrence coefficients, providing explicit formulas for their generating functions, measures, and spectral properties, and revealing connections to Chebyshev polynomials.

Contribution

It introduces a new class of orthogonal polynomials with periodic recurrences and derives explicit formulas and spectral measures, linking them to Chebyshev polynomials.

Findings

Orthogonality measure may include discrete mass points

Spectral measures are absolutely continuous

Explicit formulas for generating functions and continued fractions

Abstract

In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality measure of these polynomials, as well as the spectral measure for the associated doubly infinite tridiagonal Jacobi matrix. Notably, while the orthogonality measure may include discrete mass points, the spectral measure(s) of the doubly infinite Jacobi matrix are absolutely continuous. Additionally, we uncover an intrinsic connection between these new orthogonal polynomials and Chebyshev polynomials through a nonlinear transformation of the polynomial variables.