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

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