Painlevé V and time-dependent Jacobi polynomials

Abstract

In this paper we study the simplest deformation on a sequence of orthogonal polynomials. This in turn induces a deformation on the moment matrix of the polynomials and associated Hankel determinant. We replace the original (or reference) weight w0(x) (supported on or subsets of ) 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 and time-dependent determinants, using Sogo's terminology. If w0 is the normal density , or the gamma density xα e−x, , α > −1, then the initial value problem of the Toda equations can be trivially solved. This is because under elementary scaling and translation the orthogonality relations reduce to the original ones. However, if w0 is the beta density (1 − x)α(1 + x)β, x ∊ [ − 1, 1], α, β > −1, the resulting ‘time-dependent’ Jacobi polynomials will again satisfy a linear second-order ode, but no longer in the Sturm–Liouville form, which is to be expected. This deformation induces an irregular singular point at infinity in addition to three regular singular points of the hypergeometric equation satisfied by the Jacobi polynomials. We will show that the coefficients of this ode, as well as the Hankel determinant, are intimately related to a particular Painlevé V. In particular we show that , where is 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 nonlinear 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 Painlevé equation. The case with α = β = −1/2 arose from a certain integrable system and this was brought to our attention by A P Veselov.