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    Rights statement: Copyright 2016 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 57, 2016 and may be found at http://dx.doi.org/10.1063/1.4963170 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.

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Kernels and point processes associated with Whittaker functions

Research output: Contribution to journalJournal articlepeer-review

Published
Article number093505
<mark>Journal publication date</mark>09/2016
<mark>Journal</mark>Journal of Mathematical Physics
Issue number9
Volume57
Number of pages18
Publication StatusPublished
Early online date26/09/16
<mark>Original language</mark>English

Abstract

This article considers Whittaker's confluent hypergeometric function $W_{\kappa ,\mu }$
where $\kappa$ is real and $\mu$ is real or purely imaginary. Then
$\varphi (x)=x^{-\mu
-1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linear
system with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. The
Hankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix with
respect to the Laguerre basis and gives the Hankel matrix of moments of a
Jacobi weight $w_0(x)=x^b(1-x)^a$. The operation of translating $\varphi$ is equivalent to deforming $w_0$ to give $w_t (x)=e^{-t/x}x^b(1-x)^a$. The determinant of the Hankel matrix
of moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e's
transcendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise to
the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211
(2000), 335--358). Whittaker kernels are closely related to systems of orthogonal polynomials
for a Pollaczek--Jacobi type weight lying outside the usual Szeg\"o class.\par

Bibliographic note

Copyright 2016 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 57, 2016 and may be found at http://dx.doi.org/10.1063/1.4963170 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. This is a substantially revised version of paper with same authors and title which was previously place on AvXiv and Lancaster University Pure repositories.