Given a sequence of monic orthogonal polynomials (MOPS), {Pn}, with respect
to a quasi-definite linear functional u, we find necessary and sufficient conditions on the parameters
an and bn
for the sequence
Pn(x)+anPn−1(x)+bnPn−2(x), n≥1P0(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.