Orthogonal polynomials associated with an inverse quadratic spectral transform

Let {Pn}n≥0{Pn}n≥0 be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional uu and {Qn}n≥0{Qn}n≥0 a sequence of polynomials defined by Qn(x)=Pn(x)+snPn−1(x)+tnPn−2(x),n≥1, with tn≠0tn≠0 for n≥2n≥2.We obtain a new characterization of the orthogonality of the sequence {Qn}n≥0{Qn}n≥0 with respect to a linear functional vv, in terms of the coefficients of a quadratic polynomial hh such that h(x)v=uh(x)v=u.We also study some cases in which the parameters snsn and tntn 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{Pn}n≥0 and {Qn}n≥0{Qn}n≥0 is presented.