Kernels and point processes associated with Whittaker functions

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 $L^2(1\mathrm{/}2,\mathrm{\infty })$ and input and output spaces $\mathbf{C}$. The Hankel operator ${\mathrm{\Gamma }}_{\phi }$ on $L^2(0,\mathrm{\infty })$ 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 $\phi$ 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 $\sigma$ form of Painlev'e’s transcendental differential equation $PV$. It is shown that ${\mathrm{\Gamma }}_{\phi }$ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski (Comm. Math. Phys. 211 (2000), 335–358).\par