Kernels and point processes associated with Whittaker functions.

This article considers Whittaker's confluent hypergeometric function Wκ,μ where κ is real and μ is real or purely imaginary. Then ϕ(x) = x−μ−1/2Wκ,μ(x) arises as the scattering function of a continuous time linear system with state space L²(1/2, ∞) and input and output spaces C. The Hankel operator Γϕ on L²(0, ∞) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight w0(x) = xb(1 − x)a. The operation of translating ϕ is equivalent to deforming w0 to give wt(x) = e−t/xxb(1 − x)a. The determinant of the Hankel matrix of moments of wε satisfies the σ form of Painlevé's transcendental differential equation PV. It is shown that Γϕ 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. [ABSTRACT FROM AUTHOR]