Orthogonal polynomials associated with an inverse quadratic spectral transform

2009/09/03 by Manuel Alfaro, M. Alfaro, F. Marcellan +9
Computer Science · Mathematics · Physics and Astronomy · #33C45 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #Quantum Mechanics and Non-Hermitian Physics #math.CA #msc:33C45 #msc:42C05

paper · pdf · doi:10.48550/arxiv.0909.0619

21 pages

arxiv created 2009/09/03 · openalex publication_date 2009/09/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \P_n \_n≥0 be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional u and \Q_n \_n≥0 a sequence of polynomials defined by Q_n(x)=P_n(x)+s_n P_n-1(x)+t_n P_n-2(x), n≥1, with t_n \not= 0 for n≥2. We obtain a new characterization of the orthogonality of the sequence \Q_n \_n≥0 with respect to a linear functional v, in terms of the coefficients of a quadratic polynomial h such that h(x)v= u. We also study some cases in which the parameters s_n and t_n can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with \P_n \_n≥0 and \Q_n \_n≥0 is presented.

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