This article considers Whittaker's confluent hypergeometric function W, where κ is real and μ is real or purely imaginary. Then φ(x) = xW(x) arises as the scattering function of a continuous time linear system with state space L(1/2,∞) and input and output spaces C. The Hankel operator γ on L(0,8) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight w(x) = x(1 - x). The operation of translating φ is equivalent to deforming w to give w(x) = ex(1 - x). The determinant of the Hankel matrix of moments of w satisfies the σ form of Painlevé's transcendental differential equation PV. It is shown that γ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski [Commun. Math. Phys. 211, 335-358 (2000)]. Whittaker kernels are closely related to systems of orthogonal polynomials for a Pollaczek-Jacobi type weight lying outside the usual Szegö class.