Krall-type orthogonal polynomials and integrable isomonodromic deformations

Abstract

Krall-type polynomials are orthogonal polynomials for a Stieltjes' measure obtained by adding jumps at the boundary of the orthogonality interval of either the generalized Laguerre polynomials or the Jacobi polynomials. We show that both the recurrence relations and the second-order linear differential equations defining these polynomials are explicitly determined in terms of specific solutions of certain integrable systems. When only one jump is present, this leads to integrable cases of the Painlevé III or the Painlevé V equation. In the case of two jumps, first studied by Koornwinder, we obtain a new integrable system of partial differential equations of Schlesinger type. When the jumps are equal and the starting polynomials are the Gegenbauer polynomials, this system reduces to an integrable case of the Painlevé V equation.

39 pages. Appendix merged into Introduction and Remark 4.1. Proposition 2.1 and acknowledgements added. Notations, presentation, and references improved