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arXiv · 2412.08166

Orthogonal polynomials with periodic recurrence coefficients

Dan Dai·Mourad E. H. Ismail·Xiang-Sheng Wang
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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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Dan Dai, Mourad E. H. Ismail, Xiang-Sheng Wang. 2025-06-28. Orthogonal polynomials with periodic recurrence coefficients. https://arxiv.org/abs/2412.08166

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