From hyperbolic to complex Euler integrals

Abstract: Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

• The paper rigorously justifies the limit from hyperbolic beta integrals to two-dimensional complex Euler integrals with explicit parameter correspondence.
• It establishes sharp uniform estimates and asymptotic bounds for hyperbolic gamma functions to ensure precise control of contour deformations and error terms.
• The work extends its methods to hyperbolic conical and Ruijsenaars' integrals, highlighting applications in quantum integrable systems and representation theory.

From Hyperbolic to Complex Euler Integrals: Technical Summary and Implications

Introduction and Motivation

The paper "From hyperbolic to complex Euler integrals" (2604.04471) investigates the degeneration of univariate hyperbolic beta integrals and associated special functions to their complex hypergeometric counterparts. The principal focus concerns the rigorous limiting transition from univariate integrals over products of hyperbolic gamma functions—so-called hyperbolic hypergeometric integrals—to two-dimensional complex Euler-type integrals, culminating in explicit complex beta and conical integrals. The work draws heavily on asymptotic analysis, sharp uniform estimates for the gamma functions, and a systematic approach to limiting procedures in special function theory.

This work is situated within the program of exploring systematically the web of connections and degenerations among multivariate hypergeometric integrals of elliptic, hyperbolic, trigonometric, and rational types, extending the extensive literature on scalar-valued (single gamma function) limits [Ra, SS], and continuing—by explicit, technical means—the path from the classical Euler beta and Gauss hypergeometric functions to their more exotic, higher-level analogues.

Preliminaries: Gamma Functions and Hypergeometric Integrals

The analysis begins with a recapitulation of complex and hyperbolic gamma functions, their difference equations, reflection symmetries, and limiting relations to the classical gamma function. The complex field version, $\bm{\Gamma}(a|a')$, is defined through a Barnes-type contour integral, ensuring single-valuedness and convergence through the appropriate selection of parameter restrictions. For the hyperbolic gamma function, $\gamma^{(2)}(z; \omega_1, \omega_2)$, the authors summarize analytic continuations, difference equations, reflection identities, and the careful asymptotics underpinning their boundary behavior.

The Euler beta integral and its generalization to the complex field,

$$\int_{\mathbb{C}} [z]^{a-1}[1-z]^{b-1} d^2 z = \frac{\bm{\Gamma}(a|a') \bm{\Gamma}(b|b')}{\bm{\Gamma}(a+b|a'+b')},$$

are set in parallel to the univariate hyperbolic beta integral involving the hyperbolic gamma function and its natural region of convergence and analytic continuation. Explicit reduction formulas of hyperbolic to trigonometric to rational cases are recovered as limiting instances.

Rigorous Limit: From Hyperbolic to Complex Beta Integral

The main technical advance of the paper is the rigorous justification of the transition from univariate hyperbolic beta integrals to two-dimensional complex Euler integrals. The core result is as follows: starting with the univariate integral

$$\int_{i\mathbb{R}} e^{2\pi i \lambda z / (\omega_1 \omega_2)} \, \gamma^{(2)}(\pm z + \frac{\omega_1 + \omega_2}{2} - g) \frac{dz}{i\sqrt{\omega_1 \omega_2}},$$

in precise regimes where $\omega_1/\omega_2 \to -1$, and employing an explicit parametrization of all variables, the authors show this integral converges to

$\int_{\mathbb{C}} [t]^{a-1}[1-t]^{b-1} d^2 t,$

with matching, explicit correspondence for parameters $(a,a',b,b')$ determined by limits of the original variables. The proof leverages:

• Uniform bounds for ratios of hyperbolic gamma functions, established via sharp estimates for $q$-products and integral representations, even at or near pinched contours caused by coalescing poles.
• Contour deformation and partitioning: As the hyperbolic structure degenerates, the original contour sum is systematically decomposed into near-singular and regular parts, with rigorous error bounds showing how (improper) Riemann sums converge to the two-dimensional complex integral.
• Dominated convergence arguments and careful order-of-limits manipulations, ensuring the validity of exchanging summation, integration, and limiting procedures essential for precise asymptotics.

A centerpiece is the explicit control of tails in the limiting sums (Propositions and Lemmas in the appendices), ensuring that only local neighborhoods of the pinching points contribute in the limit and all other contributions are exponentially suppressed.

Extension: Degeneration of Hyperbolic Conical and Hypergeometric Integrals

The authors extend the degeneration procedure to the hyperbolic conical functions (closely related to Ruijsenaars' eigenfunctions for relativistic Calogero-Sutherland models) and higher-level hyperbolic hypergeometric integrals. Principal findings are:

• The hyperbolic conical function, given initially as a univariate hyperbolic gamma integral with a parameter shift, under the same double-scaling limit, degenerates to a complex rational Euler-type integral, generalizing the complex beta case.
• For the hyperbolic eight-parameter Ruijsenaars' hypergeometric function, under a suitable simultaneous scaling of all parameters, one recovers explicit complex analogues of Euler-type hypergeometric functions over the complex field.

These extensions are achieved with essentially the same uniform asymptotic and contour deformation techniques as in the beta integral case, but require more intricate parameter tracking and analysis of singularity pinching in higher codimension.

Technical Refinements and Uniform Estimates

The appendices constitute a systematic technical contribution:

• They develop a machinery of estimates for ratios and products of basic hypergeometric ($q$-) products, hyperbolic gamma functions, and their limiting behavior, which is essential for establishing the dominated convergence and tail estimations rigorously.
• Key lemmas establish error bounds on the difference between the renormalized hyperbolic gamma product and its complex limit, depending on the scaling and the region in parameter space.
• Careful control of improper Riemann sum approximations in the presence of singularities is provided, insuring that the formal interchange of sum and limit is justified for the class of integrals considered.

Implications and Outlook

Mathematical Implications

This work solidifies the precise relationships and transition rules between hyperbolic-type and complex-type hypergeometric integrals at the level of carefully regulated limiting processes. It fills a notable gap in the rigorous asymptotic analysis of nontrivial Barnes-type integrals, exposing the structure by which higher-level special functions reduce to complex analogues. The methods applied reveal the necessity of explicit control of poles, contours, and asymptotics, particularly in cases relevant for harmonic analysis, representation theory, and integrable systems (e.g., Ruijsenaars functions, modular double quantum groups).

Potential Applications and Theoretical Outlook

• These results provide a rigorous foundation for interpreting representations of quantum groups and the modular double in terms of limits to complex Lie group harmonic analysis, as the complex hypergeometric functions correspond to representation matrix elements of $SL(2,\mathbb{C})$ and its higher rank analogues.
• The techniques open the door to the systematic analysis of multidimensional hyperbolic eigenfunctions (e.g., Hallnäs–Ruijsenaars functions), and their reductions to complex analogues of Heckman–Opdam functions, bridging domains in integrable systems, special functions, and non-commutative harmonic analysis.
• The explicit identification of rigorous limiting procedures suggests a program in which holomorphic asymptotic analysis is extended beyond the lower rank, to multivariate, multicomponent, and matrix-valued situations, with direct implications for quantum integrability.

Conclusion

The paper provides an explicit technical bridge from hyperbolic hypergeometric integrals—prototypical objects in the modern special function and quantum integrable systems theory—to complex Euler-type integrals. The rigorous asymptotic methods and uniform estimates developed constitute a fundamental toolkit for specialists investigating deep connections among hypergeometric special functions, quantum group representations, and the analytic continuation structure of integrable models. The work further motivates systematic analysis of multi-dimensional analogues and corresponding spectral problems in the context of complex Lie groups and quantum harmonic analysis (2604.04471).