TY - JOUR

T1 - Kernels and point processes associated with Whittaker functions

AU - Blower, Gordon

AU - Chen, Yang

N1 - 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.

PY - 2016/9

Y1 - 2016/9

N2 - 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 linearsystem with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. TheHankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix withrespect to the Laguerre basis and gives the Hankel matrix of moments of aJacobi 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 matrixof moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e'stranscendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise tothe 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

AB - 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 linearsystem with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. TheHankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix withrespect to the Laguerre basis and gives the Hankel matrix of moments of aJacobi 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 matrixof moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e'stranscendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise tothe 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

KW - Hankel determinants

KW - Painleve differential equations

KW - random matrices

U2 - 10.1063/1.4963170

DO - 10.1063/1.4963170

M3 - Journal article

VL - 57

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 093505

ER -