169 research outputs found
The importance of the Selberg integral
It has been remarked that a fair measure of the impact of Atle Selberg's work
is the number of mathematical terms which bear his name. One of these is the
Selberg integral, an n-dimensional generalization of the Euler beta integral.
We trace its sudden rise to prominence, initiated by a question to Selberg from
Enrico Bombieri, more than thirty years after publication. In quick succession
the Selberg integral was used to prove an outstanding conjecture in random
matrix theory, and cases of the Macdonald conjectures. It further initiated the
study of q-analogues, which in turn enriched the Macdonald conjectures. We
review these developments and proceed to exhibit the sustained prominence of
the Selberg integral, evidenced by its central role in random matrix theory,
Calogero-Sutherland quantum many body systems, Knizhnik-Zamolodchikov
equations, and multivariable orthogonal polynomial theory.Comment: 43 page
Hyperbolic beta integrals
Hyperbolic beta integrals are analogues of Euler's beta integral in which the
role of Euler's gamma function is taken over by Ruijsenaars' hyperbolic gamma
function. They may be viewed as -bibasic analogues of the
beta integral in which the two bases and are interrelated
by modular inversion, and they entail -analogues of the beta integral for
. The integrals under consideration are the hyperbolic analogues of the
Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman
integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal
limit case of Spiridonov's elliptic Nassrallah-Rahman integral.Comment: 35 pages. Remarks and references to recent new developments are
added. To appear in Adv. Mat
An expansion formula for the Askey-Wilson function
The Askey-Wilson function transform is a q-analogue of the Jacobi function
transform with kernel given by an explicit non-polynomial eigenfunction of the
Askey-Wilson second order q-difference operator. The kernel is called the
Askey-Wilson function. In this paper an explicit expansion formula for the
Askey-Wilson function in terms of Askey-Wilson polynomials is proven. With this
expansion formula at hand, the image under the Askey-Wilson function transform
of an Askey-Wilson polynomial multiplied by an analogue of the Gaussian is
computed explicitly. As a special case of these formulas a q-analogue (in one
variable) of the Macdonald-Mehta integral is obtained, for which also two
alternative, direct proofs are presented.Comment: 24 pages. Some remarks added in section 6 on the connection with
moment problem
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