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

## 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) |
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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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