Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)2149-2167
Journal / PublicationJournal of Approximation Theory
Volume162
Issue number12
Publication statusPublished - Dec 2010

Abstract

We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights w(x){colon equals}w(x,t)=e-t/xxα(1-x)β,t≥0, defined for x∈[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 Painlevé V and/or allied functions.It is also shown that the logarithmic derivative of the Hankel determinant, Dn(t):=det(∫01xi+je-t/xxα(1-x)βdx)i,j=0n-1, satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new. © 2010.

Citation Format(s)

Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials. / Chen, Yang; Dai, Dan.
In: Journal of Approximation Theory, Vol. 162, No. 12, 12.2010, p. 2149-2167.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review