Orthogonal polynomials associated with an inverse quadratic spectral transform

Let {P-n}(n >= 0) be a sequence of monic orthogonal polynomials with respect to a quasi-definite linear functional u and (Q(n)}(n >= 0) a sequence of polynomials defined by Q(n)(x) = P-n(x) + s(n) P(n-1()x) + t(n) P(n-2()x), n >= 1, with t(n) not equal 0 for n >= 2. We obtain a new characterization of the orthogonality of the sequence {Q}(n >= 0) with respect to a linear functional nu, in terms of the coefficients of a quadratic polynomial h such that h(x) upsilon = u. We also study some cases in which the parameters s(n) and t(n) can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with {P-n}(n >= 0) and {Q(n))(n >= 0) is presented. (C) 2010 Elsevier Ltd. All rights reserved.