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/content/aip/journal/jmp/57/9/10.1063/1.4963170
2016-09-26
2016-10-22

Abstract

This article considers Whittaker’s confluent hypergeometric function where is real and is real or purely imaginary. Then () = −1/2 () arises as the scattering function of a continuous time linear system with state space 2(1/2, ∞) and input and output spaces . The Hankel operator Γ on 2(0, ∞) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight () = (1 − ). The operation of translating is equivalent to deforming to give () = / (1 − ). The determinant of the Hankel matrix of moments of satisfies the form of Painlevé’s transcendental differential equation . It is shown that Γ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski [Commun. Math. Phys. , 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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