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

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Given {Pₙ } a sequence of monic orthogonal polynomials, we analyze their linear combinations {Qₙ }with constant coefficients and fixed length k+1. Necessary and sufficient conditions are given for the orthogonality of the monic sequence {Qₙ } as well as an interesting interpretation in terms of the Jacobi matrices associated with {Pₙ } and {Qₙ }. Moreover, in the case k=2, we characterize the families {Pₙ } such that the corresponding polynomials {Qₙ } are also orthogonal.