Continued fractions and orthogonal polynomials on the unit circle

Abstract

This survey is written to stress the role of continued fractions in the theory of orthogonal polynomials on the line and on the circle. We follow the historical development of the subject, which opens many interesting relationships of orthogonal polynomials to other important branches of mathematics. At the end we present a new formula for orthogonal polynomials on the real line, the Leganés formula, ∫ Q n - 1 2 d σ t - z = 1 Q n / Q n - 1 - ∫ d σ n / ( t - z ) , which is a correct analogue of the corresponding formula on the unit circle. This formula is applied to obtain a recent result by Simon.