Convex linear combinations of sequences of monic orthogonal polynomials

For a sequence $\{\Phi _n\}_0^\infty$ of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sufficient conditions on a SMOP $\{Q_n\}_0^\infty$ in order that a convex linear combination $\{R_n\}_0^\infty$ with $R_n=\beta \Phi _n+(1-\beta)Q_n$ be a SMOP with respect to a positive measure supported on the unit circle.