Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

Mourad E. H. Ismail, Nicholas S. Witte

Published 2000-12-29, updated 2001-08-01Version 2

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.