Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

Abstract

AbstractOur work studies sequences of orthogonal polynomials{Pn(x)}n⩾0of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parametert. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations intfor the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type PVI. The particular case of PVIarising here has the same four parameters as the solution found by Magnus (1995J. Comput. Appl. Math.57215–37) but differs in the boundary conditions.