a Department of Mathematics, University of South Florida, Tampa, Florida, 33620-5700, U.S.A.;b Department of Mathematics and Statistics and School of Physics, University of Melbourne, Victoria, 3010, Australia
Abstract:
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.