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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 continuoustime linear system with state space $L^2(1/2, \infty )$ and input and outputspaces ${\bf C}$. The Hankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ isexpressed as a matrix with respect to the Laguerre basis and gives the Hankelmatrix 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 transcendentaldifferential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise to theWhittaker kernel from random matrix theory, as studied by Borodin and Olshanski(Comm. Math. Phys. 211 (2000), 335--358).\par

Kernels and point processes associated with Whittaker functions