Painlev\'e V and a Pollaczek-Jacobi type orthogonal polynomials

TL;DR

This paper investigates a family of orthogonal polynomials with a parameter-dependent weight, revealing their connection to Painlevé V equations and deriving new nonlinear difference equations for related determinants.

Contribution

It introduces a novel analysis of orthogonal polynomials with a singular exponential weight, linking their recurrence coefficients to Painlevé V and deriving a new nonlinear difference equation for the Hankel determinant.

Findings

Recurrence coefficients expressed via Painlevé V functions.

Hankel determinant's logarithmic derivative satisfies Painlevé V σ-form.

A new second-order nonlinear difference equation for the determinant.

Abstract

We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients. For $t>0,$ the factor $\rex^{-t/x}$ induces an infinitely strong zero at $x=0.$ With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are a particular Painlev\'e V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, $$ D_n(t):=\det(\int_{0}^{1} x^{i+j} \:\rex^{-t/x}\:x^{\al}(1-x)^{\bt}dx)_{i,j=0}^{n-1}, $$ satisfies the Jimbo-Miwa-Okamoto $\sigma-$form of the Painlev\'e V and that the same quantity satisfies a second order non-linear difference equation which we believe to be new.