Some Macdonald-Mehta Integrals by Brute Force

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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 F4 case of the second generalization. As well we give a two parameter generalization for the dihedral group H2 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 two-parameter version for H6 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.