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### Electronic data

• JMPWhittakerrevised

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

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Published
Article number 093505 09/2016 Journal of Mathematical Physics 9 57 18 Published 26/09/16 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.