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By using the Szeg\H{o}'s transformation we deduce new relations between therecurrence coefficients for orthogonal polynomials on the real line and theVerblunsky parameters of orthogonal polynomials on the unit circle. Moreover,we study the relation between the corresponding $\mathcal{S}$-functions and$\mathcal{C}$-functions

On perturbed orthogonal polynomials on the real line and the unit circle via Szeg\H{o}'s transformation