Painlev\'e V and time dependent Jacobi polynomials
Abstract
In this paper we study the simplest deformation on a sequence of orthogonal polynomials, namely, replacing the original (or reference) weight w0(x) defined on an interval by w0(x)e-tx. It is a well-known fact that under such a deformation the recurrence coefficients denoted as αn and βn 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 zn-1 of the monic orthogonal polynomials associated with the "time-dependent" Jacobi weight, satisfies, up to a translation in t, the Jimbo-Miwa σ-form of the same PV; while a recurrence coefficient αn(t), is up to a translation in t and a linear fractional transformation PV(α2/2,-β2/2, 2n+1+α+β,-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.