Let $\{ P_n \}_{ n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{ Q_n \}_{ n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{ n-1}(x)+t_n P_{ n-2}(x),\quad n\ge1,$$ with $t_n \not= 0$ for $n\ge2$. We obtain a new characterization of the orthogonality of the sequence $\{ Q_n \}_{ n\ge0}$ 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 $\{ P_n \}_{ n\ge0}$ and $\{ Q_n \}_{ n\ge0}$ is presented.
Categories: math.CA