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 interval of orthogonality 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 some integrable systems. When there is only one jump, we are led to integrable cases of the Painlev\'e III or the Painlev\'e 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 Painlev\'e V equation.