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

Given { P n } n ≥ 0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e.,  Q n ( x ) = P n ( x ) + a 1 P n − 1 ( x ) + ⋯ + a k P n − k , a k ≠ 0 , n > k . Necessary and sufficient conditions are given for the orthogonality of the sequence { Q n } n ≥ 0 . An interesting interpretation in terms of the Jacobi matrices associated with { P n } n ≥ 0 and { Q n } n ≥ 0 is shown. er, in the case k = 2 , we characterize the families { P n } n ≥ 0 such that the corresponding polynomials { Q n } n ≥ 0 are also orthogonal.