Kernels and point processes associated with Whittaker functions

Abstract. This article considers Whittaker’s confluent hypergeometric function Wκ,μ where κ is real and μ is real or purely imaginary. Then φ(x) = xWκ,μ(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 Γφ on L(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) = x (1 − x). The operation of translating φ is equivalent to deforming w0 to give wt(x) = e−t/xxb(1 − x). 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 (Comm. Math. Phys. 211 (2000), 335–358). Whittaker kernels are closely related to systems of orthogonal polynomials for a Pollaczek–Jacobi type weight lying outside the usual Szegö class.