Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials

TL;DR

This paper develops integrable differential systems for deformed Laguerre-Hahn orthogonal polynomials, linking their recurrence coefficients to Painlevé VI equations and providing new insights into their differential properties.

Contribution

It introduces a novel framework connecting Laguerre-Hahn orthogonal polynomials with integrable systems and Painlevé equations, including explicit Lax pairs and differential equations for deformation parameters.

Findings

Derived differential equations for recurrence coefficients

Established Lax pairs for the polynomial systems

Connected special cases to Painlevé VI equations

Abstract

Our work studies sequences of orthogonal polynomials $ \{P_{n}(x)\}_{n=0}^{\infty} $ of the Laguerre-Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, are subject to a deformation parameter $t$. We derive systems of differential equations and give Lax pairs, yielding non-linear differential equations in $t$ for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a M\"{o}bius transformation of a Stieltjes function related to a modified Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlev\'e type P$_\textrm{VI}$. The particular case of P$_\textrm{VI}$ arising here has the same four parameters as the solution found by Magnus [A.P. Magnus, Painlev\'e-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comput. Appl. Math., 57:215-237, 1995] but differs in the boundary conditions.