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Kernels and point processes associated withWhittaker functions
Gordon Blower1; Yang Chen2
2016-09
Source PublicationJournal of Mathematical Physics
ISSN0022-2488
Volume57Issue:9
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) = xμ−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 w0to give wt(x) = et/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.

KeywordHankel Determinants Painlev ́e Differential Equations
DOIhttps://doi.org/10.1063/1.4963170
Language英语
WOS Research AreaPhysics
WOS SubjectPhysics, Mathematical
WOS IDWOS:000385564900032
PublisherAMER INST PHYSICS, 1305 WALT WHITMAN RD, STE 300, MELVILLE, NY 11747-4501 USA
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Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorGordon Blower
Affiliation1.Department of Mathematics and Statistics, Lancaster University, LA1 4YF, England, UK
2.Department of Mathematics, University of Macau, Avenda da Universidade, Taipa, Macau,China
Recommended Citation
GB/T 7714
Gordon Blower,Yang Chen. Kernels and point processes associated withWhittaker functions[J]. Journal of Mathematical Physics,2016,57(9).
APA Gordon Blower,&Yang Chen.(2016).Kernels and point processes associated withWhittaker functions.Journal of Mathematical Physics,57(9).
MLA Gordon Blower,et al."Kernels and point processes associated withWhittaker functions".Journal of Mathematical Physics 57.9(2016).
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