Painlev\'e V and time dependent Jacobi polynomials

TL;DR

This paper explores how time-dependent Jacobi polynomials relate to Painlevé V equations, revealing deep connections between orthogonal polynomial deformations, nonlinear differential equations, and Fredholm determinants.

Contribution

It establishes a novel link between time-dependent Jacobi polynomials and Painlevé V, using Toda equations and difference equations to connect orthogonal polynomials to integrable systems.

Findings

Coefficients of the associated ODE relate to Painlevé V.

The monic orthogonal polynomial coefficients satisfy Jimbo-Miwa σ-form of Painlevé V.

A Fredholm determinant connected to Toeplitz plus Hankel operators is linked to Painlevé equations.

Abstract

In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight $w_0(x)$ defined on an interval by $w_0(x)e^{-tx}.$ It is a well-known fact that under such a deformation the recurrence coefficients denoted as $\alpha_n$ and $\beta_n$ evolve in $t$ according to the Toda equations, giving rise to the time dependent orthogonal polynomials, using Sogo's terminology. The resulting "time-dependent" Jacobi polynomials satisfy a linear second order ode. We will show that the coefficients of this ode are intimately related to a particular Painlev\'e V. In addition, we show that the coefficient of $z^{n-1}$ of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in $t,$ the Jimbo-Miwa $\sigma$-form of the same $P_{V};$ while a recurrence coefficient $\alpha_n(t),$ is up to a translation in $t$ and a linear fractional transformation $P_{V}(\alpha^2/2,-\beta^2/2, 2n+1+\alpha+\beta,-1/2).$ These results are found from combining a pair of non-linear difference equations and a pair of Toda equations. This will in turn allow us to show that a certain Fredholm determinant related to a class of Toeplitz plus Hankel operators has a connection to a Painlev\'e equation.