Krall-type orthogonal polynomials and integrable isomonodromic deformations

TL;DR

This paper explores Krall-type orthogonal polynomials with added boundary jumps, linking their properties to integrable systems like Painlevé equations and Schlesinger systems, revealing new connections and integrable cases.

Contribution

It explicitly determines recurrence relations and differential equations for Krall-type polynomials using integrable system solutions, including new cases related to Painlevé and Schlesinger equations.

Findings

Recurrence relations linked to integrable systems.

New integrable PDE system of Schlesinger type.

Special cases reduce to Painlevé V equation.

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.