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

Given {Pn}n≥0{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≠0,n>k. Necessary and sufficient conditions are given for the orthogonality of the sequence {Qn}n≥0{Qn}n≥0. An interesting interpretation in terms of the Jacobi matrices associated with {Pn}n≥0{Pn}n≥0 and {Qn}n≥0{Qn}n≥0 is shown.Moreover, in the case k=2k=2, we characterize the families {Pn}n≥0{Pn}n≥0 such that the corresponding polynomials {Qn}n≥0{Qn}n≥0 are also orthogonal.