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Orthogonal polynomials generated by a linear structure relation: inverse problem. (English) Zbl 1264.33011
Summary: Let \((P_{n})_{n}\) and \((Q_{n})_{n}\) be two sequences of monic polynomials linked by a type of structure relation such as
\[ Q_{n}(x)+r_{n}Q_{n-1}(x)=P_{n}(x)+s_{n}P_{n-1}(x)+t_{n}P_{n-2}(x) \]
where \((r_n)_n\), \((s_n)_n\) and \((t_n)_n\) are sequences of complex numbers.
First, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences \((P_n)_n\) and \((Q_n)_n\) are orthogonal with respect to regular moment linear functionals \(\mathbf u\) and \(\mathbf v\), respectively.
Second, assuming that the above relation is non-degenerate and \((P_n)_n\) is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence \((Q_n)_n\) in terms of the coefficients of polynomials \(\varPhi\) and \(\varPsi\) which appear in the rational transformation (in the distributional sense) \(\varPhi \mathbf u=\varPsi \mathbf v\).
Some illustrative examples for the developed theory are presented.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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