Limits of elliptic hypergeometric integrals
Eric M. Rains
TL;DR
The paper formalizes and unifies the limiting degenerations of multivariate elliptic hypergeometric integrals, establishing uniform asymptotics for the generalized gamma functions to justify hyperbolic, trigonometric, rational, and classical limits directly from the elliptic level. It proves that these limits converge exponentially fast with explicit error factors, and derives new trigonometric identities and Weyl-group invariances (notably $E_7$ and $E_8$-type structures) arising in the limiting processes. The approach hinges on a suite of tools—uniform asymptotics for $\Gamma_h^{(r)}$, $\Gamma_e^{(r)}$, and $\Gamma_t$, generalized triangle inequalities, and tail-exchange localization arguments—to localize integrands and control convergence across multiple limit regimes. By connecting elliptic-level transformations to their hyperbolic, trigonometric, rational, and classical counterparts, the work clarifies the hierarchy of multivariate special functions and sets the stage for deriving low-level results as limits of elliptic identities, including limits of biorthogonal and interpolation functions. It also extends known univariate results to rich multivariate analogues, broadening the landscape of exact evaluation and transformation formulas in elliptic and degenerate regimes.
Abstract
In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.