| In 1944 Atle Selberg published an n-dimensional generalization of Euler's beta function integral, and in 1962 Freeman Dyson published a conjecture, later proven by Gunson and Wilson, that also arises from an n-dimensional integral. I. G. Macdonald noticed that both formulas could be "indexed" by the finite root systems of type BC and A, respectively. Macdonald also noted that a generalization of Dyson's formula, first conjectured by George Andrews in 1975 (and as yet proven only in low dimension), could be indexed in a similar manner by the so-called affine root systems of type A.; We give here two conjectures. Each evaluates the constant term of a certain Laurent polynomial derived from a reduced, irreducible afine root system. Conjecture A as applied to the affine root systems S(BC(,n)) generalizes Selberg's version of Euler's integral. Applied to other systems, in particular the exceptional systems of type E, F, and G, it gives new (but yet unproven) generalizations of the beta function. Conjecture B contains Andrews' conjecture and two isolated formulas for affine root systems S(A(,2)) and S(B(,2)). The latter two formulas are proven here. We also show that another application of Conjecture B to the affine systems S(A(,n)) is in a limiting case equivalent to Selberg's generalization of Cauchy's "beta" integral on a non-compact interval.; We prove certain low-dimensional cases of these conjectures using F. H. Jackson's sum of a particular well-poised basic hypergeometric series. The higher dimensional versions of the conjectures thus imply multiple-sum generalizations of Jackson's formula. |
