Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials
Seminar Room 1, Newton Institute
All classical orthogonal polynomials in the Askey table (and its q-extension) are orthogonal with respect to a weight w that satisfies a first order differential, (divided) difference or q-difference equation with polynomial coefficients of degree one and less than or equal to two respectively (Pearson equation). Their recurrence coefficients coefficients can be found using this Pearson equation by solving a first order difference equation. Semi-classical weights satisfy a Pearson equation with polynomial coefficients of higher degree. The recurrence coefficients then satisfy higher order difference equations. Many examples have been worked out and we will present some of the semi-classical weights that give rise to discrete Painlevé equations for the recurrence coefficients of the orthogonal polynomials. This is joint work with my student Lies Boelen.