The paper introduces an analogue of a theorem of Szegő on orthogonal polynomials on the unit circle. Let \(\left\{ a_n\right\}_{n=0}^\infty\) be a sequence of complex numbers such that \[ |a_n |<1,\; \forall n.\tag{1} \] A monic polynomial \(\Phi_{n+1}\) can be defined using the recurrence relations \[ \Phi_{n+1}(z)=z\Phi_{n}(z)-\overline{a}_n \Phi^*_{n}(z); \quad \Phi_0(z)=1 , \]
\[ \Phi^*_{n+1}(z)=\Phi^*_{n}(z) -{a}_n z \Phi_{n}(z) , \] where \( \Phi^*_{n}(z)=z^n \overline{\Phi_n}(1/\overline{z})\).
The sequence of complex numbers satisfying (1) is called Verblunsky coefficients. For any such sequence, there exists a probability measure \(\mu\) for which \(\Phi_n\) is the \(n\)th monic orthogonal polynomial with respect to \(\mu\) on the unit circle \(\mathbb{T}\). Condition (1) is necessary and sufficient for the existence of a Schur function \(f\) which is analytic in the unit disc \(\mathbb{D}\) and maps \(\mathbb{D}\) into its closure with \(\sup_{z\in \mathbb{D}}\left| f(z)\right|\leq 1 .\) The Schur function gives rise to a Carathéodory function \(F(z)\) given by \[ F(z)=\frac{ 1+zf(z)}{1-zf(z)} . \] If we decompose the measure \[ d\mu=w(\theta) \frac{d\theta}{2\pi} + d\mu_s, \] where \(w\in L^1\left(\mathbf{T}, \frac{d\theta}{2\pi}\right)\) and \(d\mu_s\) is the singular part of the measure, it can be shown that \[ \operatorname{Re} F(e^{i\theta}) =\frac{1-\left|f(e^{i\theta})\right|^2 }{ \left|1- e^{i\theta} f(e^{i\theta})\right|^2 =w(\theta)}. \] Szegő’s theorem relates the Verblunsky coefficients to the measure via the relation \[ \prod_{j=0}^\infty \left( 1-|a_j|^2\right)= \exp \left( \int_0^{2\pi}\log \left( \operatorname{Re} F(e^{i\theta})\right) \frac{d\theta}{2\pi} \right). \]
In the paper under review, the authors replace condition (1) with the following condition which implies that the first \(N\) coefficients lie outside the closed unit disc \[ |a_n|\neq 1, n=0,1, \cdots , N-1, \text{ and } |a_n|<1, \quad \forall n \geq N, \] and prove the following analogue of Szegő’s theorem \[ \prod_{j=0}^\infty \left( 1-|a_j|^2\right)= \prod_{j=0}^m |\lambda_j|^{-2} \exp \left( \int_0^{2\pi}\log \left( \operatorname{Re} F(e^{i\theta})\right) \frac{d\theta}{2\pi} \right) , \] where \(\lambda_j, \; j=0, 1, \cdots , m,\) are the poles in \(\mathbb{D}\) of a function \(F\) that plays the role of the Carathéodory function. An analogue of a theorem of Verblunsky is also derived.