Generating new classes of orthogonal polynomials

Given a sequence of monic orthogonal polynomials (MOPS), { P n } , with respect to a quasi-definite linear functional u , we find necessary and sufficient conditions on the parameters a n and b n for the sequence P n ( x ) + a n P n − 1 ( x ) + b n P n − 2 ( x ) , n ≥ 1 P 0 ( x ) = 1 , P − 1 ( x ) = 0 to be orthogonal. In particular, we can find explicitly the linear functional v such that the new sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite and Tchebychev orthogonal polynomials of second kind are obtained. We also solve a problem of this type for orthogonal polynomials with respect to a Hermitian linear functional.