- JMPWhittakerrevised
**Rights statement:**Copyright 2016 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 57, 2016 and may be found at http://dx.doi.org/10.1063/1.4963170 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.Accepted author manuscript, 271 KB, PDF document

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- http://scitation.aip.org/content/aip/journal/jmp/57/9/10.1063/1.4963170
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In: Journal of Mathematical Physics, Vol. 57, No. 9, 093505, 09.2016.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Blower, G & Chen, Y 2016, 'Kernels and point processes associated with Whittaker functions', *Journal of Mathematical Physics*, vol. 57, no. 9, 093505. https://doi.org/10.1063/1.4963170

Blower, G., & Chen, Y. (2016). Kernels and point processes associated with Whittaker functions. *Journal of Mathematical Physics*, *57*(9), Article 093505. https://doi.org/10.1063/1.4963170

Blower G, Chen Y. Kernels and point processes associated with Whittaker functions. Journal of Mathematical Physics. 2016 Sept;57(9):093505. Epub 2016 Sept 26. doi: 10.1063/1.4963170

@article{bebc07d5654e4cadb19a3d5fe01604d9,

title = "Kernels and point processes associated with Whittaker functions",

abstract = "This article considers Whittaker's confluent hypergeometric function $W_{\kappa ,\mu }$where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu-1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linearsystem with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. TheHankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix withrespect to the Laguerre basis and gives the Hankel matrix of moments of aJacobi weight $w_0(x)=x^b(1-x)^a$. The operation of translating $\varphi$ is equivalent to deforming $w_0$ to give $w_t (x)=e^{-t/x}x^b(1-x)^a$. The determinant of the Hankel matrixof moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e'stranscendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise tothe 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 Szeg\{"}o class.\par",

keywords = "Hankel determinants, Painleve differential equations, random matrices",

author = "Gordon Blower and Yang Chen",

note = "Copyright 2016 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 57, 2016 and may be found at http://dx.doi.org/10.1063/1.4963170 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. This is a substantially revised version of paper with same authors and title which was previously place on AvXiv and Lancaster University Pure repositories.",

year = "2016",

month = sep,

doi = "10.1063/1.4963170",

language = "English",

volume = "57",

journal = "Journal of Mathematical Physics",

issn = "0022-2488",

publisher = "American Institute of Physics Publising LLC",

number = "9",

}

TY - JOUR

T1 - Kernels and point processes associated with Whittaker functions

AU - Blower, Gordon

AU - Chen, Yang

N1 - Copyright 2016 American Institute of Physics. The following article appeared in Journal of Mathematical Physics, 57, 2016 and may be found at http://dx.doi.org/10.1063/1.4963170 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. This is a substantially revised version of paper with same authors and title which was previously place on AvXiv and Lancaster University Pure repositories.

PY - 2016/9

Y1 - 2016/9

N2 - This article considers Whittaker's confluent hypergeometric function $W_{\kappa ,\mu }$where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu-1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linearsystem with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. TheHankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix withrespect to the Laguerre basis and gives the Hankel matrix of moments of aJacobi weight $w_0(x)=x^b(1-x)^a$. The operation of translating $\varphi$ is equivalent to deforming $w_0$ to give $w_t (x)=e^{-t/x}x^b(1-x)^a$. The determinant of the Hankel matrixof moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e'stranscendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise tothe 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 Szeg\"o class.\par

AB - This article considers Whittaker's confluent hypergeometric function $W_{\kappa ,\mu }$where $\kappa$ is real and $\mu$ is real or purely imaginary. Then $\varphi (x)=x^{-\mu-1/2}W_{\kappa ,\mu }(x)$ arises as the scattering function of a continuous time linearsystem with state space $L^2(1/2, \infty )$ and input and output spaces ${\bf C}$. TheHankel operator $\Gamma_\varphi$ on $L^2(0, \infty )$ is expressed as a matrix withrespect to the Laguerre basis and gives the Hankel matrix of moments of aJacobi weight $w_0(x)=x^b(1-x)^a$. The operation of translating $\varphi$ is equivalent to deforming $w_0$ to give $w_t (x)=e^{-t/x}x^b(1-x)^a$. The determinant of the Hankel matrixof moments of $w_\varepsilon$ satisfies the $\sigma$ form of Painlev\'e'stranscendental differential equation $PV$. It is shown that $\Gamma_\varphi$ gives rise tothe 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 Szeg\"o class.\par

KW - Hankel determinants

KW - Painleve differential equations

KW - random matrices

U2 - 10.1063/1.4963170

DO - 10.1063/1.4963170

M3 - Journal article

VL - 57

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 093505

ER -