A general explicit first and second order differential equation, plus a Verblunsky-coefficient difference equation, is established for semiclassical orthogonal polynomials on the unit circle with Pearson weights of degree at most three.
Generalized semiclassical orthogonal polynomials on the unit circle: A Riemann-Hilbert perspective
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abstract
In this work we show how to get advantage from the Riemann--Hilbert analysis in order to obtain first and second order differential equations for the orthogonal polynomials and associated functions with a weight on the unit circle. We deduce properties for the recurrence relation coefficients from differential properties of the weight. We take the so called generalized modified Jacobi and Bessel weights as a case study.
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Differential equations for a class of semiclassical orthogonal polynomials on the unit circle
A general explicit first and second order differential equation, plus a Verblunsky-coefficient difference equation, is established for semiclassical orthogonal polynomials on the unit circle with Pearson weights of degree at most three.