[Paper Review] Limits of elliptic hypergeometric integrals

Eric M. Rains|ArXiv.org|Jul 4, 2006

Mathematical functions and polynomials16 references4 citations

TL;DR

This paper establishes rigorous limits of multivariate elliptic hypergeometric integrals by deriving uniform asymptotic estimates for generalized gamma functions, proving that hyperbolic, trigonometric, rational, and classical integrals arise as well-defined limits from the elliptic level. It provides new trigonometric integral identities and confirms that hyperbolic integrals are not formal degenerations but actual limiting cases with exponentially small error bounds.

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.

Motivation & Objective

• To rigorously establish that hyperbolic, trigonometric, rational, and classical hypergeometric integrals are actual limiting cases of multivariate elliptic hypergeometric integrals, rather than formal degenerations.
• To resolve the gap in prior work where hyperbolic integrals were derived via degeneration of proofs rather than limits of the integrals themselves.
• To derive new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level, extending known results.
• To provide strong, uniform asymptotic estimates for multiple gamma functions to support the limit analysis across all levels.
• To clarify the relationship between elliptic integrals and lower-level hypergeometric integrals, enabling new results at each level via limiting procedures.

Proposed method

• Derives uniform asymptotic estimates for generalized elliptic and hyperbolic gamma functions, extending to multiple gamma functions of arbitrary order.
• Applies a master inequality (Lemma 3.2) derived from asymptotic analysis of an elliptic analogue of the Cauchy determinant to identify where integrands are maximized.
• Uses a tail-exchange argument to rigorously justify the convergence of integrals in the hyperbolic and rational limits, with exponentially small error bounds.
• Breaks symmetry in the integrand when necessary to ensure both sides of transformation identities have well-defined limits in the trigonometric and classical cases.
• Applies uniform estimates to handle degenerations from the elliptic level directly to each lower level (hyperbolic, trigonometric, rational, classical), avoiding intermediate steps.
• Uses the $q \to 1$ and $p \to 0$ limits to derive classical and rational integrals, with careful analysis of parameter constraints and convergence conditions.

Experimental results

Research questions

• RQ1 Can hyperbolic hypergeometric integrals be rigorously derived as limits of elliptic hypergeometric integrals, rather than by degenerating the proof?
• RQ2 What new trigonometric integral identities emerge as limits from the elliptic level, particularly when symmetry is broken in the integrand?
• RQ3 How do the asymptotics of generalized gamma functions support the convergence of integrals in the rational and classical limits?
• RQ4 What conditions ensure that the classical limit yields a non-degenerate integral, and how do the parameters affect the resulting measure?
• RQ5 Can the orthogonality of Macdonald polynomials be recovered as a limiting case of biorthogonal abelian functions in the elliptic setting?

Key findings

• The hyperbolic integrals of van Diejen and Spiridonov are rigorously shown to be limits of the elliptic integrals, with exponentially small error estimates.
• New trigonometric integral identities are derived as limits from the elliptic level, including cases where both sides of a transformation have well-defined limits upon symmetry breaking.
• The Macdonald polynomial orthogonality relations are recovered as limiting cases of biorthogonal abelian functions, confirming the 'conjectures' of Macdonald as limits of elliptic identities.
• The rational limit is obtained by taking $q \to 1$ with $p$ fixed, requiring symmetry breaking and a tail-exchange argument, yielding a hybrid of hyperbolic and trigonometric behavior.
• The classical limit is characterized by exponential decay of the integrand unless certain inequalities on arguments are satisfied, leading to a multivariate beta-type integral with explicit evaluation.
• Corollary 7.5 provides a new evaluation of a classical multivariate integral with power and rational weight functions, valid under weaker conditions than previously known.