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For a sequence fng1 0 of monic orthogonal polynomials (SMOP), with respect to a positive measure supported on the unit circle, we obtain necessary and sucient conditions on a SMOP fQng1 0 in order that a convex linear combination fRng1 0 with Rn = n +( 1 ) Q n be a SMOP with respect to a positive measure supported on the unit circle. n(0)n+1(z )=( z n(0) + n+1(0)) n(z) z(1 j n(0)j 2 )n+1(0)n 1(z); n 0, with initial conditions 1(z )=0a nd 0 ( z )=1 : The values n(0) are called reflection parameters, and they satisfy jn(0)j < 1

Convex linear combinations of sequences of monic orthogonal polynomials