When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Given {Pn}n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., Qn(x) = Pn(x) + a1Pn−1(x) + · · ·+ akPn−k, ak 6= 0, n > k. Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0 and {Qn}n≥0 is shown. Moreover, in the case k = 2, we characterize the families {Pn}n≥0 such that the corresponding polynomials {Qn}n≥0 are also orthogonal. MSC 2000 :33C45, 42C05