Kernels and point processes associated with Whittaker functions

Abstract

This article considers Whittaker's function $W_{κ,μ}$ where $κ$ is real and $μ$ is real or purely imaginary. Then $φ(x)=x^{-μ-1/2}W_{κ,μ}(x)$ 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 $Γ_φ$ 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 $φ$ 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 $σ$ form of Painlevé's transcendental differential equation $PV$. It is shown that $Γ_φ$ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211 (2000), 335--358).\par