Let { Φ n } be a monic orthogonal polynomial sequence on the unit circle. We define recursively a new sequence { Ψ n } of polynomials by the following linear combination: Ψ n ( z ) + p n Ψ n - 1 ( z ) = Φ n ( z ) + q n Φ n - 1 ( z ) , p n , q n ∈ ℂ , p n q n ≠ 0 . In this paper, we give necessary and sufficient conditions in order to make { Ψ n } be an orthogonal polynomial sequence too. Moreover, we obtain an explicit representation for the Verblunsky coefficients { Φ n ( 0 ) } and { Ψ n ( 0 ) } in terms of p n and q n . Finally, we show the relation between their corresponding Carathéodory functions and their associated linear functionals.