An elliptic hypergeometric integral with W(F_4) symmetry

TL;DR

This paper introduces a novel elliptic hypergeometric integral transformation with $W(F_4)$ symmetry, generalizing the Langer–Schlosser–Warnaar construction and extending the elliptic beta framework. The main result is a transformation asserting the $W(F_4)$-invariance of the elliptic hypergeometric integral $E(b;t;p,q)$, together with an explicit integral identity that recovers the LSW case in a suitable limit. By studying $p\to0$ limits, the authors derive a rich family of basic hypergeometric objects with preserved $W(F_4)$ symmetry, including a basic integral $B_1$, the $B_2$ and ${}_{14}W_{13}$ representations, and multiple ${}_{k}\phi_{l}$ and Nassrallah–Rahman–type expressions on various faces of a 24-cell polytope. These results illuminate how elliptic symmetries degenerate to structured basic hypergeometric identities and yield new integral and series representations, notably for ${}_{14}W_{13}$ and ${}_8W_7$ cases, thereby enriching the landscape of multivariate theta- and gamma-function identities.

Abstract

In this article we give a new transformation between elliptic hypergeometric beta integrals, which gives rise to a Weyl group symmetry of type F_4. The transformation is a generalization of a series transformation discovered by Langer, Schlosser, and Warnaar. Moreover we consider various limits of this transformation to basic hypergeometric functions obtained by letting p tend to 0.

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Definition 4.1
• Proposition 4.2