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Transformations of elliptic hypergometric integrals

Eric M. Rains

Abstract

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, 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 E_7.

Transformations of elliptic hypergometric integrals

Abstract

We prove a pair of transformations relating elliptic hypergeometric integrals of different dimensions, corresponding to the root systems BC_n and A_n; as a special case, we recover some integral identities conjectured by van Diejen and Spiridonov. For BC_n, 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 E_7.

Paper Structure

This paper contains 10 sections, 73 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. The Anderson integral
  3. The $BC_n\leftrightarrow BC_m$ transformation
  4. The $A_n\leftrightarrow A_m$ transformation
  5. Mixed transformations
  6. Difference operators
  7. Integral operators
  8. Biorthogonal functions
  9. Type II transformations
  10. Appendix: Meromorphy of integrals

Key Result

Theorem 2.1

AndersonGW:1991 Let $x_1,\dots x_n$ and $s_1,\dots s_n$ be sequences of real numbers such that Then where $S=\sum_{1\le i\le n}s_i$.

Theorems & Definitions (162)

  • Definition 1
  • Theorem 2.1
  • Remark
  • Theorem 2.2
  • Remark
  • Remark
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 152 more