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Inversions of integral operators and elliptic beta integrals on root systems

Vyacheslav P. Spiridonov, S. Ole Warnaar

TL;DR

This work develops a unified inversion framework for elliptic beta integrals tied to root systems, first establishing a univariate inversion via contour deformations and a kernel $\Delta(z,w;t)$, then extending to the multivariate root-system pairs $(\mathrm{A}_n,\mathrm{A}_n)$ and $(\mathrm{A}_n,\mathrm{C}_n)$. The authors construct explicit inversion kernels, prove two main inversion theorems, and leverage them to derive a novel type I $\mathrm{A}_n$ elliptic beta integral by inverting a $\mathrm{C}_n$ integral, with an independent $q$-difference proof provided in a separate section. Consequences include new integral identities and elliptic hypergeometric series sums (notably a multivariate $D_n$-type sum), enriching the theory of integral Bailey pairs for elliptic hypergeometric functions. The results deepen the symmetry and transform structure of elliptic beta integrals on root systems and connect to known limiting cases such as Gustafson’s $q$-beta integrals. Overall, the paper broadens both the methodological toolkit and the catalog of exact evaluation formulas in elliptic hypergeometric theory.

Abstract

We prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems. Using the (A,C) inversion formula to invert one of the known C-type elliptic beta integrals, we obtain a new elliptic beta integral for the root system of type A. Validity of this integral is established by a different method as well.

Inversions of integral operators and elliptic beta integrals on root systems

TL;DR

This work develops a unified inversion framework for elliptic beta integrals tied to root systems, first establishing a univariate inversion via contour deformations and a kernel , then extending to the multivariate root-system pairs and . The authors construct explicit inversion kernels, prove two main inversion theorems, and leverage them to derive a novel type I elliptic beta integral by inverting a integral, with an independent -difference proof provided in a separate section. Consequences include new integral identities and elliptic hypergeometric series sums (notably a multivariate -type sum), enriching the theory of integral Bailey pairs for elliptic hypergeometric functions. The results deepen the symmetry and transform structure of elliptic beta integrals on root systems and connect to known limiting cases such as Gustafson’s -beta integrals. Overall, the paper broadens both the methodological toolkit and the catalog of exact evaluation formulas in elliptic hypergeometric theory.

Abstract

We prove a novel type of inversion formula for elliptic hypergeometric integrals associated to a pair of root systems. Using the (A,C) inversion formula to invert one of the known C-type elliptic beta integrals, we obtain a new elliptic beta integral for the root system of type A. Validity of this integral is established by a different method as well.

Paper Structure

This paper contains 14 sections, 14 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Elliptic beta integrals
  4. Inversion formulas. I. The single variable case
  5. Motivation
  6. The $n=1$ integral inversion
  7. Proof of Theorem \ref{['inversionthm']}
  8. Inversion formulas. II. The root systems A$_n$ and C$_n$
  9. Main results
  10. Consequences of Theorems \ref{['inversionthmA']} and \ref{['inversionthmAC']}
  11. Integral identities
  12. Series identities
  13. Proof of Theorem \ref{['inversionthmAC']}
  14. Alternative proof of the new A$_n$ elliptic beta integral

Key Result

Theorem 4.1

Let $p,q,t\in\mathbb C$ such that $M<\lvert t\rvert^2<1$. For fixed $w\in\mathbb T$ let $C_w$ denote a contour inside the annulus $\mathbb A=\{z\in\mathbb C|~ \lvert t\rvert-\epsilon<\lvert z\rvert<\lvert t\rvert^{-1}+\epsilon\}$ for infinitesimally small but positive $\epsilon$, such that $C_w$ has where

Theorems & Definitions (21)

  • Theorem 4.1
  • Theorem 5.1: (A,A) inversion formula
  • Theorem 5.2: (A,C) inversion formula
  • Theorem 5.3
  • Theorem 5.4
  • Conjecture 5.1: (C,A) inversion formula
  • Theorem 5.5
  • Theorem 5.6
  • Proposition 5.1
  • Lemma 5.1
  • ...and 11 more