Kernels and point processes associated with Whittaker functions

Blower, Gordon and Chen, Yang (2016) Kernels and point processes associated with Whittaker functions. Journal of Mathematical Physics, 57 (9). ISSN 0022-2488

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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

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Journal of Mathematical Physics
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Copyright 2016 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 57, 2016 and may be found at 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.
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14 Sep 2016 08:12
Last Modified:
26 Sep 2020 04:14