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Orthogonal polynomials associated with an inverse quadratic spectral transform. (English) Zbl 1217.42050
Summary: Let \(\{P_{n}\}_{n\geq 0}\) be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional \(u\) and \(\{Q_{n}\}_{n\geq 0}\) a sequence of polynomials defined by \(Q_n(x)=P_n(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x),~ n\geq 1\) with \(t_{n}\neq 0\) for \(n\geq 2\). We obtain a new characterization of the orthogonality of the sequence \(\{Q_{n}\}_{n\geq 0}\) 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 \(\{P_{n}\}_{n\geq 0}\) and \(\{Q_{n}\}_{n\geq 0}\) is presented.

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
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