Integrable Differential Systems for Deformed Laguerre-Hahn Orthogonal Polynomials

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 Möbius 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 Painlevé type P${}_{\text{VI}}$. The particular case of P${}_{\text{VI}}$ arising here has the same four parameters as the solution found by Magnus [A.P. Magnus, Painlevé-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.

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