Orthogonal polynomials associated with an inverse quadratic spectral transform

Let {Pn}n≥0 be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional u and {Qn}n≥0 a sequence of polynomials defined by Qn(x) = Pn(x) + sn Pn−1(x) + tn Pn−2(x), n ≥ 1, with tn ̸= 0 for n ≥ 2. We obtain a new characterization of the orthogonality of the sequence {Qn}n≥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 sn and tn can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 is presented.