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On the generalised Selberg integral of Richards and Zheng · Around

2007

On the generalised Selberg integral of Richards and Zheng

Warnaar, S. Ole

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In a recent paper Richards and Zheng compute the determinant of a matrix whose entries are given by beta-type integrals, thereby generalising an earlier result by Dixon and Varchenko.

They then use their result to obtain a generalisation of the famous Selberg integral.
In this note we point out that the Selberg-generalisation of Richards and Zheng is a special case of an integral over Jack polynomials due to Kadell.
We then show how an integral formula for Jack polynomials of Okounkov and Olshanski may be applied to prove Kadell's integral along the lines of Richards and Zheng.

Built on

A. L. Dixon, Generalisations of Legendre’s formula K E ′ − ( K − E ) K ′ = 1 2 π KE^{\prime}-(K-E)K^{\prime}=\frac{1}{2}\pi , Proc. London Math. Soc. (2) 3 (1905), 206–224
1905
Earlier work this paper cites.
A. Selberg, Bemerkninger om et multipelt integral, Norske Mat. Tidsskr. 26 (1944), 71–78
1944
Earlier work this paper cites.
A. Varchenko, The Euler beta-function, the Vandermonde determinant, the Legendre equation, and critical values of linear functions on a configuration of hyperplanes. I. Izv. Akad. Nauk. SSSR Ser. Mat. 53 (1989), 1206–1235
Original
1989
Earlier work this paper cites.

Similar

G. W. Anderson, A short proof of Selberg’s generalized beta formula, Forum Math. 3 (1991), 415–417
1991
Cited alongside, same era.
I. G. Macdonald, Symmetric functions and Hall polynomials, second edition, (Oxford University Press, New York, 1995)
1995
Cited alongside, same era.
K. W. J. Kadell, The Selberg–Jack symmetric functions, Adv. Math. 130 (1997), 33–102
1997
Cited alongside, same era.
L. Euler, De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt, Comm. Acad. Sci. Petropolitanae 5 (1730), 36–57
Cited in the paper.
D. Richards, Q. Zheng, The determinant of a hypergeometric period matrix and a generalization of Selberg’s integral, to appear in Adv. Appl. Math
Cited in the paper.

Then

A. Okounkov, (Shifted) Macdonald polynomials: q q -integral representation and combinatorial formula, Compositio Math. 112 (1998), 147–182
1998
Later among the works it cites.
D. Richards, Q. Zheng, Determinants of period matrices, and an application to Selberg’s multidimensional beta integral, Adv. Appl. Math. 28 (2002), 602–633
2002
Later among the works it cites.

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