  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 $L^2(1/2, \infty )$ 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 $w$. The operation of translating
$\varphi$ 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 $\sigma$ 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).\par
