Some MacdonaldMehta Integrals by Brute Force
 Frank G. Garvan
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Abstract
Bombieri and Selberg showed how Mehta’s [6; p. 42] integral could be evaluated using Selberg’s [7] integral. Macdonald [5; §§5,6] conjectured two different generalizations of Mehta’s integral formula. The first generalization is in terms of finite Coxeter groups and depends on one parameter. The second generalization is in terms of root systems and the number of parameters in equal to the number of different root lengths. In the case of Weyl groups Macdonald showed how the first generalization follows from the second. We give a proof of the 13 case of the first generalization and the F_{4} case of the second generalization. As well we give a two parameter generalization for the dihedral group H_{2} ^{2n}. The parameters are constant on each of the two orbits. We note that the G _{2} case of the second generalization follows from our twoparameter version for H^{6} _{2}. Our proofs draw on ideas from AomotO’s [1] proof of Selberg’s integral and Zeilberger’s [10] proof of the G ^{v} _{2} case of the Macdonald Morris [5; Conj. 3.3] constant term root system conjecture. The problem is reduced to solving a system of linear equations. These equations were generated and solved by the computer algebra package MAPLE.
 K. Aomoto, Jacobi polynomials associated with Seiberg’s integral, SIAM J. Math. Anal., 18 (1987), 545–549. CrossRef
 F.G. Garvan, A beta integral associated with the root system G _{2}, SIAM J. Math. Anal., 19 (1988) (to appear).
 F.G. Garvan, A proof of the MacdonaldMorris root system conjecture for F _{4}, submitted.
 L.C. Grove AND C.T. Benson, Finite Reflection Groups, SpringerVerlag, New York, 1985.
 I.G. Macdonald, Some conjectures for root systems and finite reflection groups, SIAM J. of Math. Anal., 13 (1982), 988–1007. CrossRef
 M.L. Mehta, Random Matrices, Academic Press, New York and London, 1967.
 A. Selberg, Bermerkninger om et multiplet integral, Norske Mat. Tidsskr, 26 (1944), 71–78.
 J.R. Stembridge, A short proof of Macdonald’s conjecture for the root system of type A, Proc. Amer. Math. Soc., 102 (1988), 777–786.
 E.C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.
 E.T. Whittaker AND G.N. Watson, A Course in Modern Analysis, Cambridge University Press, Cambridge, 1927.
 D. Zeilberger, A unified approach to Macdonald’s rootsystem conjectures, SIAM J. Math. Anal., 19 (1988), (to appear).
 Title
 Some MacdonaldMehta Integrals by Brute Force
 Book Title
 qSeries and Partitions
 Pages
 pp 7798
 Copyright
 1989
 DOI
 10.1007/9781468406375_8
 Print ISBN
 9781468406399
 Online ISBN
 9781468406375
 Series Title
 The IMA Volumes in Mathematics and Its Applications
 Series Volume
 18
 Series ISSN
 09406573
 Publisher
 Springer US
 Copyright Holder
 SpringerVerlag New York
 Additional Links
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 Editors

 Dennis Stanton ^{(1)}
 Editor Affiliations

 1. School of Mathematics, University of Minnesota
 Authors

 Frank G. Garvan ^{(2)}
 Author Affiliations

 2. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, Minnesota, 55455, USA
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