Kernels and point processes associated with Whittaker functions

Abstract

This article considers Whittaker's function $W_{\hat{I}º,\hat{I}¼}$ where $\hat{I}º$ is real and $\hat{I}¼$ is real or purely imaginary. Then $\ddot{I}†(x)=x^{-\hat{I}¼-1/2}{W}_{\hat{I}º,\hat{I}¼}(x)$ arises as the scattering function of a continuous time linear system with state space $L^2(1/2,\infty )$ and input and output spaces $\mathbf{C}$. The Hankel operator $\hat{I}{“}_{\ddot{I}}†$ 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 $\ddot{I}†$ is equivalent to multiplying $w$ by an exponential factor to give $w_{\epsilon }$. The determinant of the Hankel matrix of moments of $w_{\epsilon }$ satisfies the $\ddot{I}ƒ$ form of Painlevé's transcendental differential equation $PV$. It is shown that $\hat{I}{“}_{\ddot{I}}†$ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211 (2000), 335--358).\par

19 pages