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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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