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Kernels and point processes associated with Whittaker functions
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This article considers Whittaker’s confluent hypergeometric function W
κ,μ where κ is real and μ is real or purely imaginary. Then φ(x) = x
κ,μ(x) arises as the scattering function of a continuous time linear system with state space L
2(1/2, ∞) and input and output spaces C. The Hankel operator Γφ on L
2(0, ∞) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight w
0(x) = xb(1 − x)a. The operation of translating φ is equivalent to deforming w
0 to give wt(x) = e
xb(1 − x)a. The determinant of the Hankel matrix of moments of w
ε 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.
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