Kernels and point processes associated with Whittaker functions

Blower, Gordon and Chen, Yang (2015) Kernels and point processes associated with Whittaker functions. Working Paper. UNSPECIFIED.

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Abstract

This article considers Whittaker's 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 $w$. The operation of translating $\varphi$ is equivalent to multiplying $w$ by an exponential factor to give $w_\varepsilon$. 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).

Item Type: Monograph (Working Paper)
Uncontrolled Keywords: Painleve equations ; random matrices ; Kankel determinants
Departments: Faculty of Science and Technology > Mathematics and Statistics
ID Code: 76214
Deposited By: ep_importer_pure
Deposited On: 21 Oct 2015 05:11
Refereed?: No
Published?: Published
Last Modified: 11 Jun 2019 02:40
URI: https://eprints.lancs.ac.uk/id/eprint/76214

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