Continued fractions and orthogonal polynomials on the unit circle
This survey paper contains the notes of a series of lectures given by the author at Caltech. As indicated in the abstract, the use of continued fractions in the theory of orthogonal polynomials on the unit circle is highlighted. The paper touches on the usual subjects, including the connection between periodicity and quadratic function fields. Interesting is the result on the development of a Cauchy integral into a continued fraction. This straightforward generalization of the corresponding formula for the unit circle to the real line case is dubbed the `Leganés formula' (Theorem 26) and runs as follows. Let \(\{Q_k\}_{k\geq 1}\) be the denominators for the continued fraction \[ \int \frac {d\sigma} {t-z} =\frac{1}{b_1}{{}\atop{-}}\frac{1}{b_2}{{}\atop{-\cdots-}}\frac{1}{b_n-\int d\sigma_n/(t-z)}. \] with \[ b_k(z)=-{z\over m_k}+\ell_k,\;0<m_k = \int\, d\sigma_{k-1}, \;\ell_k= \int\,td\sigma_{k-1}, \] and \(\sigma\) a measure with finite moments on \textbf{R}. Then \[ \int\,{Q_{n-1}^2 d\sigma\over t-z} = {1\over Q_n/Q_{n-1} - \int\,d\sigma_n /(t-z)}. \]
scientific article; zbMATH DE number 2157674
scientific article