Orthogonal polynomials generated by a linear structure relation: Inverse problem

Let ( P n ) n and ( Q n ) n be two sequences of monic polynomials linked by a type 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. 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 u and v , respectively. , 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 the polynomials Φ and Ψ which appear in the rational transformation (in the distributional sense) Φ u = Ψ v . llustrative examples of the developed theory are presented.