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 w0 to 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.

1.
Arazy
,
J.
, “
More on convergence in unitary matrix spaces
,”
Proc. Am. Math. Soc.
83
,
44
48
(
1981
).
2.
Basor
,
E.
,
Chen
,
Y.
, and
Ehrhardt
,
T.
, “
Painlevé V and time-dependent Jacobi polynomials
,”
J. Phys. A: Math. Theor.
43
,
015204
(
2010
).
3.
Blower
,
G.
, “
Hankel operators that commute with second-order differential operators
,”
J. Math. Anal. Appl.
342
,
601
614
(
2008
).
4.
Blower
,
G.
, “
Integrable operators and the squares of Hankel operators
,”
J. Math. Anal. Appl.
340
,
943
953
(
2008
).
5.
Blower
,
G.
, “
Linear systems and determinantal point processes
,”
J. Math. Anal. Appl.
355
,
311
334
(
2009
).
6.
Borodin
,
A.
and
Olshanski
,
G.
, “
Point processes and the infinite symmetric group
,”
Math. Res. Lett.
5
,
799
816
(
1998
).
7.
Borodin
,
A.
and
Olshanski
,
G.
, “
Distributions on partitions, point processes and the hypergeometric kernel
,”
Commun. Math. Phys.
211
,
335
358
(
2000
).
8.
Borodin
,
A.
and
Olshanski
,
G.
, “
Infinite random matrices and ergodic measures
,”
Commun. Math. Phys.
223
,
87
123
(
2001
).
9.
Borodin
,
A.
and
Soshnikov
,
A.
, “
Janossy densities. I. Determinantal ensembles
,”
J. Stat. Phys.
113
,
595
610
(
2003
).
10.
Chen
,
M.
and
Chen
,
Y.
, “
Singular linear statistics of the Laguerre unitary ensemble and Painlevé III: Double scaling analysis
,”
J. Math. Phys.
56
,
063506
(
2015
).
11.
Chen
,
Y.
and
Dai
,
D.
, “
Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials
,”
J. Approximation Theory
162
,
2149
2167
(
2010
).
12.
Chen
,
Y.
and
Zhang
,
L.
, “
Painlevé VI and the unitary Jacobi ensembles
,”
Stud. Appl. Math.
125
,
91
112
(
2010
).
13.
Daley
,
D. J.
and
Vere-Jones
,
D.
,
An Introduction to the Theory of Point Processes
(
Springer-Verlag
,
New York
,
1958
).
14.
Dyson
,
F. J.
, “
Correlations between eigenvalues of a random matrix
,”
Commun. Math. Phys.
19
,
235
250
(
1970
).
15.
Erdélyi
,
A.
,
Higher Transcendental Functions
(
McGraw–Hill
,
New York
,
1954
), Vol.
1
.
16.
Erdélyi
,
A.
,
Tables of Integral Transforms
(
McGraw–Hill
,
New York
,
1954
), Vol.
II
.
17.
Gradsteyn
,
I. S.
and
Ryzhik
,
I. M.
,
Table of Integrals, Series and Products
, 7th ed. (
Elsevier/Academic Press
,
Amsterdam
,
2007
).
18.
Gromak
,
V. I.
,
Laine
,
I.
, and
Shimomura
,
S.
,
Painlevé Differential Equations in the Complex Plane
(
Walter de Gruyter
,
2002
).
19.
Howland
,
J. S.
, “
Spectral theory of operators of Hankel type I
,”
Indiana Univ. Math. J.
41
,
409
426
(
1992
).
20.
Jimbo
,
M.
, “
Monodromy problem and the boundary condition for some Painlevé equations
,”
Publ. Res. Inst. Math. Sci.
18
,
1137
1161
(
1982
).
21.
Jimbo
,
M.
and
Miwa
,
T.
, “
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II
,”
Physica D
2
,
407
448
(
1981
).
22.
Jimbo
,
M.
,
Miwa
,
T.
, and
Ueno
,
K.
, “
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and τ-function
,”
Physica D
2
,
306
352
(
1981
).
23.
Lang
,
S.
,
SL2(R)
(
Springer
,
New York
,
1985
).
24.
Lisovyy
,
O.
, “
Dyson’s constant for the hypergeometric kernel
,” in
New Trends in Quantum Integrable Systems
(
World Scientific Publishing
,
Hackensack NJ
,
2011
), pp.
243
267
.
25.
Martínez-Finkelshtein
,
A.
and
Sanchéz-Lara
,
J. F.
, “
Shannon entropy of symmetric Pollaczek polynomials
,”
J. Approximation Theory
145
,
55
80
(
2007
).
26.
Mehta
,
M. L.
,
Random Matrices
, 3rd ed. (
Elsevier
,
San Diego
,
2004
).
27.
Okamoto
,
K.
, “
On the τ-function of the Painlevé equations
,”
Physica D
2
,
525
535
(
1981
).
28.
Olshanski
,
G.
, “
Point processes and the infinite symmetric group, Part V: Analysis of the matrix Whittaker kernel
,” e-print arXiv:9810014.
29.
Peller
,
V. V.
,
Hankel Operators and Their Applications
(
Springer–Verlag
,
New York
,
2003
).
30.
Rosenblum
,
M.
, “
On the Hilbert matrix I
,”
Proc. Am. Math. Soc.
9
,
137
140
(
1958
).
31.
Sneddon
,
I. N.
,
The Use of Integral Transforms
(
McGraw–Hill
,
1972
).
32.
Soshnikov
,
A.
, “
Determinantal random point fields
,”
Russ. Math. Surv.
55
,
923
975
(
2000
).
33.
Szegö
,
G.
,
Orthogonal Polynomials
(
American Mathematical Society
,
New York
,
1939
).
34.
Tracy
,
C. A.
and
Widom
,
H.
, “
Fredholm determinants, differential equations and matrix models
,”
Commun. Math. Phys.
163
,
33
72
(
1994
).
35.
Tracy
,
C. A.
and
Widom
,
H.
, “
Level spacing distributions and the Airy kernel
,”
Commun. Math. Phys.
159
,
151
174
(
1994
).
36.
Tracy
,
C. A.
and
Widom
,
H.
, “
Level spacing distributions and the Bessel kernel
,”
Commun. Math. Phys.
161
,
289
309
(
1994
).
37.
Tracy
,
C. A.
and
Widom
,
H.
, “
Random unitary matrices, permutations and Painlevé
,”
Commun. Math. Phys.
207
,
665
685
(
1999
).
38.
Turrittin
,
H. L.
, “
Reduction of ordinary differential equations to Birkhoff canonical form
,”
Trans. Am. Math. Soc.
107
,
485
507
(
1963
).
39.
Whittaker
,
E. T.
and
Watson
,
G. N.
,
A Course of Modern Analysis
, 4th ed. (
Cambridge University Press
,
1996
).
40.
Wimp
,
J.
, “
A class of integral transforms
,”
Proc. Edinburgh Math. Soc.
14
,
33
40
(
1964
).
You do not currently have access to this content.