2003/09/16 by Eric M. Rains, Rains, Eric M. · 4 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Mathematical functions and polynomials #Nonlinear Waves and Solitons #math.CA #math.CO #math.QA
paper · pdf · doi:10.48550/arxiv.math/0309252
58 pages, LaTeX. v2: some notations changed to harmonize with sequel. v3: added proof of main conjecture; other minor changes. v4: appendix on meromorphy added
arxiv created 2005/04/20 · arxiv updated 2009/12/01
We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BCn and An; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BCn, we also consider their "Type II" integral. Their proof of that integral, together with our transformation, gives rise to pairs of adjoint integral operators; a different proof gives rise to pairs of adjoint difference operators. These allow us to construct a family of biorthogonal abelian functions generalizing the Koornwinder polynomials, and satisfying the analogues of the Macdonald conjectures. Finally, we discuss some transformations of Type II-style integrals. In particular, we find that adding two parameters to the Type II integral gives an integral invariant under an appropriate action of the Weyl group E7.