Painleve V and a Pollaczek-Jacobi type orthogonal polynomials

[摘要] We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights w(x) := w(x, t) = e(-t/x)x(alpha)(1-x)(beta), t >= 0, defined for x is an element of [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 e (-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 particular Painleve V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, Dn(t) := det (integral(1)(0) x(i)+(j)e(-t/x)x(alpha)(I - x)(beta)dx)(i,j=0)(n-1) satisfies the Jimbo-Miwa-Okamoto sigma-form of the Painleve V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new. Crown Copyright (C) 2010 Published by Elsevier Inc. All rights reserved.

[发布日期] 2010-12-01