Orthogonal polynomials associated with an inverse quadratic spectral transform

Abstract

Let $\{P_n{\}}_{n\ge 0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n{\}}_{n\ge 0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n{P}_{n-1}(x)+t_n{P}_{n-2}(x),n\ge 1,$$ with $t_n\ne 0$ for $n\ge 2$. We obtain a new characterization of the orthogonality of the sequence $\{Q_n{\}}_{n\ge 0}$ 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 $s_n$ 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 $\{P_n{\}}_{n\ge 0}$ and $\{Q_n{\}}_{n\ge 0}$ is presented.

21 pages