Orthogonal polynomials generated by a linear structure relation: Inverse problem

Let (Pn)n(Pn)n and (Qn)n(Qn)n be two sequences of monic polynomials linked by a type structure relation such as Qn(x)+rnQn−1(x)=Pn(x)+snPn−1(x)+tnPn−2(x),Qn(x)+rnQn−1(x)=Pn(x)+snPn−1(x)+tnPn−2(x), where (rn)n(rn)n, (sn)n(sn)n and (tn)n(tn)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 (Pn)n(Pn)n and (Qn)n(Qn)n are orthogonal with respect to regular moment linear functionals u and v, respectively.Second, assuming that the above relation is non-degenerate and (Pn)n(Pn)n is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence (Qn)n(Qn)n in terms of the coefficients of the polynomials ΦΦ and ΨΨ which appear in the rational transformation (in the distributional sense) Φu=Ψv.Some illustrative examples of the developed theory are presented.