Orthogonal polynomials with periodic recurrence coefficients

2024/12/11 by Dan Dai, Mourad E. H. Ismail, Dai, Dan +3
Mathematics · #30B70 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Primary 33D45 #Secondary 39A06

paper · pdf · doi:10.48550/arxiv.2412.08166

openalex publication_date 2024/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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