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Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

Research paper by Mourad E. H. Ismail, Nicholas S. Witte

Indexed on: 01 Aug '01Published on: 01 Aug '01Published in: Mathematics - Classical Analysis and ODEs



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.