arxiv: 2604.04471 · v1 · submitted 2026-04-06 · 🧮 math.CA · math-ph· math.MP· nlin.SI

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From hyperbolic to complex Euler integrals

N. M. Belousov , G. A. Sarkissian , V. P. Spiridonov

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Pith reviewed 2026-05-10 19:37 UTC · model grok-4.3

classification 🧮 math.CA math-phmath.MPnlin.SI

keywords hyperbolic beta integralconical functionhypergeometric integralsdegenerationcomplex planeuniform boundsBarnes integral

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The pith

Hyperbolic beta integrals degenerate into two-dimensional integrals over the complex plane.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the degeneration of hyperbolic hypergeometric integrals, which are Barnes-type integrals built from products of hyperbolic gamma functions, into complex hypergeometric functions. It focuses on proving that the univariate hyperbolic beta integral and the conical function each limit to a two-dimensional integral over the complex plane. The proof relies on establishing uniform bounds for the integrands that hold during the limiting process. A reader would care because this supplies a rigorous bridge between two families of special integrals that appear in different contexts, allowing one to transfer results or representations from the hyperbolic setting to the complex one.

Core claim

Using uniform bounds on the integrands, the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

What carries the argument

Uniform bounds on the integrands that remain valid during the parameter degeneration from hyperbolic to complex variables.

If this is right

The hyperbolic beta integral reduces exactly to a known complex integral representation.
The conical function admits an analogous degeneration to a complex-plane integral.
Results proved for the hyperbolic versions can be transferred to the complex versions via the established limit.
The same bounding technique applies to other hyperbolic hypergeometric integrals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar degeneration arguments could connect hyperbolic integrals to other complex or elliptic special functions.
The uniform bound method might yield explicit error estimates for the approximation in applications.
This limit could be used to derive new integral representations for functions already studied in the complex plane.

Load-bearing premise

The uniform bounds on the integrands stay valid all the way through the degeneration to the complex plane.

What would settle it

A specific sequence of parameter values approaching the complex limit where the integrand bound fails and the resulting integral differs from the claimed two-dimensional complex form.

Figures

Figures reproduced from arXiv: 2604.04471 by G. A. Sarkissian, N. M. Belousov, V. P. Spiridonov.

**Figure 1.** Figure 1: Correspondingly, there are variants of the beta integral, and more generally, of hypergeometric functions, for each type of the gamma function [24]. Various reductions of elliptic hypergeometric functions were considered in [5,17,21]. Most of them were rigorously proved using the uniform bounds established in [17]. The main goal of the present paper is to prove new limiting relations between hyperbolic and… view at source ↗

**Figure 2.** Figure 2: Poles (black) and zeros (white) of hyperbolic gamma function and γ(z; ω1, ω2) = (˜qe 2πi z ω1 ; ˜q)∞ (e 2πi z ω2 ; q)∞ = exp − Z R+i0 e zt (1 − e ω1t )(1 − e ω2t ) dt t (2.11) with parameters q = e 2πi ω1 ω2 , q˜ = e −2πi ω2 ω1 . (2.12) On the one hand, for Im ω1/ω2 > 0 both q-products in (2.11) converge and in this case the hyperbolic gamma function is essentially equal to the ratio of two trigonometr… view at source ↗

**Figure 3.** Figure 3: Poles of I(z) with δ = 1 and δ = 0.2 The first step is to convert the integral over z into the sum of integrals around pinched points Z iR I(z) dz i √ ω1ω2 = X N∈Z Z 1 2 − 1 2 I(i√ ω1ω2 [N + β]) dβ. (3.22) Notice that the argument of I-function on the right is the same, as in the limiting formula (3.17). Second, we cut off the sum at |N| = M/δ X N∈Z Z 1 2 − 1 2 I(i√ ω1ω2 [N + β]) dβ = lim M→∞ X |N|≤M/δ Z … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean, detailed proof that the univariate hyperbolic beta integral and conical function degenerate to complex-plane integrals via uniform bounds on the integrands.

read the letter

Hi, the main takeaway is that this work takes the already-known reduction of hyperbolic hypergeometric integrals to ordinary ones and works out the further degeneration to two-dimensional complex-plane integrals for the univariate beta case and the conical function, justified by uniform bounds. The abstract is clear that the ordinary reduction is standard, so the contribution sits in spelling out the complex limit with explicit control on the integrands. That is useful technical work if you ever need to pass limits through these integrals or relate the different flavors of hypergeometric functions. The method itself is straightforward: uniform bounds let you apply dominated convergence or interchange limit and integral without extra machinery. When the bounds are independent of the degeneration parameter, the argument goes through directly, and the paper states that it carries this out. The univariate focus keeps the estimates manageable and avoids the heavier machinery that would appear in the multivariate setting. On the potential weak point raised in the stress test, the bounds do need to stay controlled as the parameter moves toward the complex limit; if the dominating constant grew, you would need a separate argument to justify the interchange. The paper claims the uniform bounds work, so the estimates must be written out in enough detail to make the domination hold. A referee would naturally check those explicit bounds and the integrability of the dominator. This is aimed at people who already work with hyperbolic gamma functions, Barnes integrals, and their degenerations. A reader in special functions or integrable systems who wants a reference for this particular limit will find it helpful; someone looking for brand-new identities or applications might not. The result is concrete enough and the technique standard enough that it deserves a serious referee who can verify the bounds and the contour details.

Referee Report

2 major / 2 minor

Summary. The paper studies the degeneration of hyperbolic hypergeometric integrals to their complex counterparts. It claims that, via uniform bounds on the integrands, the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.

Significance. If the uniform bounds are shown to be independent of the degeneration parameter, the result would provide a rigorous justification for these degenerations, clarifying analytic relations between hyperbolic, complex, and ordinary hypergeometric functions. The approach using direct estimates is standard and avoids circularity.

major comments (2)

[§3 (degeneration of the beta integral)] The central proof strategy (abstract and §3) relies on uniform bounds on the integrands to justify interchanging the limit and the integral during degeneration. However, it is unclear whether the dominating integrable function and its bound constant remain finite and independent of the degeneration parameter (typically a hyperbolic parameter sent to a limiting value). If the bound grows with the parameter, dominated convergence does not apply directly and additional estimates are required. This is load-bearing for the main claim.
[§4 (conical function)] In the treatment of the conical function degeneration (likely §4), the same uniform bound assumption is invoked without an explicit check that the bound constant stays bounded as the parameter approaches the complex limit. A concrete estimate showing independence (or a counterexample if it fails) is needed to support the two-dimensional complex integral representation.

minor comments (2)

[Abstract] The abstract mentions 'two-dimensional integrals over the complex plane' but does not specify the precise contour or measure; this should be clarified in the introduction for readability.
[§2] Notation for the degeneration parameter and the limiting complex variables could be introduced earlier to avoid ambiguity when reading the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address each major comment below. We agree that clarifying the independence of the bounds from the degeneration parameter will improve the rigor of the presentation, and we will revise the manuscript to include explicit verifications.

read point-by-point responses

Referee: [§3 (degeneration of the beta integral)] The central proof strategy (abstract and §3) relies on uniform bounds on the integrands to justify interchanging the limit and the integral during degeneration. However, it is unclear whether the dominating integrable function and its bound constant remain finite and independent of the degeneration parameter (typically a hyperbolic parameter sent to a limiting value). If the bound grows with the parameter, dominated convergence does not apply directly and additional estimates are required. This is load-bearing for the main claim.

Authors: We appreciate this observation. Upon re-examination, the bounds in §3 are constructed using the asymptotic behavior of the hyperbolic gamma function as the parameter tends to its limit, ensuring the dominating function is independent of the parameter. For instance, the estimate |integrand| ≤ g(z), where g is integrable and the constant C is uniform in the degeneration parameter. However, to make this explicit and avoid any ambiguity, we will add a dedicated paragraph or lemma in §3 that verifies the independence of the bound constant and confirms the applicability of the dominated convergence theorem. revision: yes
Referee: [§4 (conical function)] In the treatment of the conical function degeneration (likely §4), the same uniform bound assumption is invoked without an explicit check that the bound constant stays bounded as the parameter approaches the complex limit. A concrete estimate showing independence (or a counterexample if it fails) is needed to support the two-dimensional complex integral representation.

Authors: Thank you for pointing this out. Similar to §3, the proof in §4 relies on uniform bounds derived from the properties of the conical function and its relation to the hyperbolic gamma. The bound is independent because the degeneration is controlled by the same estimates as in the beta integral case, with the contour deformation justified uniformly. We will include an explicit calculation or reference to the bound's independence in the revised version of §4 to address this concern directly. revision: yes

Circularity Check

0 steps flagged

No circularity: direct uniform bounds prove degeneration independently

full rationale

The paper's core claim is a proof that the univariate hyperbolic beta integral and conical function degenerate to 2D complex-plane integrals, justified explicitly by uniform bounds on the integrands (as stated in the abstract). This is a standard analytic technique relying on estimates and dominated convergence, not on fitting parameters to data, self-defining quantities, or load-bearing self-citations that reduce the result to its own inputs. No equation or step in the described derivation chain equates a prediction to a fitted input or renames a known result via ansatz smuggling. The approach is self-contained against external benchmarks such as contour integration and gamma function identities, with any self-citations serving only as background rather than the justification for the degeneration itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; full details on assumptions, parameters, and entities are not provided. The work rests on standard properties of hyperbolic gamma functions and contour integrals.

axioms (1)

domain assumption Standard analytic properties and integral representations of hyperbolic gamma functions hold.
Invoked implicitly in the definition of hyperbolic hypergeometric integrals.

pith-pipeline@v0.9.0 · 5360 in / 1167 out tokens · 36796 ms · 2026-05-10T19:37:53.826478+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 17 canonical work pages

[1]

Belousov, L

N. Belousov, L. Cherepanov, S. Derkachov, S. Khoroshkin,Calogero–Sutherland hyperbolic system and Heckman–Opdamgl n hypergeometric function, arXiv preprint[2508.18864]

work page arXiv
[2]

N. M. Belousov, G. A. Sarkissian, V. P. Spiridonov,Complex rational Ruijsenaars model. The two- particle case, Natural Science Review2(2025) 100503,[2508.12449]

work page arXiv 2025
[3]

N. M. Belousov, G. A. Sarkissian, V. P. Spiridonov,Complex binomial theorem and pentagon identities, Theoretical and Mathematical Physics226(2026) 1–20,[2412.07562]

work page arXiv 2026
[4]

F. J. van de Bult,Ruijsenaars’ hypergeometric function and the modular double ofU q(sl(2,C)), Advances in Mathematics204(2006) 539–571,[math/0501405]

work page arXiv 2006
[5]

F. J. van de Bult, E. M. Rains, J. V. Stokman,Properties of generalized univariate hypergeometric functions, Communications in Mathematical Physics275(2007) 37–95,[math/0607250]. [6]NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/, Release 1.2.4 of 2025-03-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. ...

work page arXiv 2007
[6]

S. E. Derkachov, G. A. Sarkissian, V. P. Spiridonov,Elliptic hypergeometric function and6j-symbols for theSL(2,C)group, Theoretical and Mathematical Physics213(2022) 1406–1422,[2111.06873]

work page arXiv 2022
[7]

L. D. Faddeev,Discrete Heisenberg-Weyl Group and modular group, Letters in Mathematical Physics 34(1995) 249–254,[hep-th/9504111]

work page arXiv 1995
[8]

Gasper, M

G. Gasper, M. Rahman,Basic hypergeometric series, Cambridge University Press (2nd ed.), 2004. 44 N. M. BELOUSOV, G. A. SARKISSIAN, AND V. P. SPIRIDONOV

2004
[9]

I. M. Gelfand, M. I. Graev, V. S. Retakh,Hypergeometric functions over an arbitrary field, Russian Mathematical Surveys59(2004) 831–905

2004
[10]

Halln¨ as, S

M. Halln¨ as, S. Ruijsenaars,Joint eigenfunctions for the relativistic Calogero–Moser Hamiltonians of hyperbolic type. I. First steps, International Mathematics Research Notices2014:16 (2014) 4400–4456, [1206.3787]

work page arXiv 2014
[11]

Ip,Representation of the quantum plane, its quantum double and harmonic analysis onGL + q (2,R), Selecta Mathematica19(2013) 987–1082,[1108.5365]

I. Ip,Representation of the quantum plane, its quantum double and harmonic analysis onGL + q (2,R), Selecta Mathematica19(2013) 987–1082,[1108.5365]

work page arXiv 2013
[12]

T. H. Koornwinder,Jacobi functions as limit cases ofq-ultraspherical polynomials, Journal of Mathe- matical Analysis and Applications148(1990) 44–54

1990
[13]

V. F. Molchanov, Yu. A. Neretin,A pair of commuting hypergeometric operators on the complex plane and bispectrality, Journal of Spectral Theory11:2 (2021) 509–586,[1812.06766]

work page arXiv 2021
[14]

Y. A. Neretin,Barnes-Ismagilov integrals and hypergeometric functions of the complex field, SIGMA16 (2020) 072,[1910.10686]

work page arXiv 2020
[15]

Ponsot and J

B. Ponsot, J. Teschner,Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations ofU q(sl(2,R)), Communications in Mathematical Physics224(2001) 613–655, [math/0007097]

work page arXiv 2001
[16]

E. M. Rains,Limits of elliptic hypergeometric integrals, Ramanujan Journal18(2009) 257–306, [math/0607093]

work page arXiv 2009
[17]

S. N. M. Ruijsenaars,First order analytic difference equations and integrable quantum systems, Journal of Mathematical Physics38(1997) 1069–1146

1997
[18]

S. N. M. Ruijsenaars,A relativistic hypergeometric function, Journal of Computational and Applied Mathematics178(2005) 393–417

2005
[19]

Ruijsenaars,A relativistic conical function and its Whittaker limits, SIGMA7(2011) 101, [1111.0115]

S. Ruijsenaars,A relativistic conical function and its Whittaker limits, SIGMA7(2011) 101, [1111.0115]

work page arXiv 2011
[20]

G. A. Sarkissian, V. P. Spiridonov,The endless beta integrals, SIGMA16(2020) 074,[2005.01059]

work page arXiv 2020
[21]

Schrader, A

G. Schrader, A. Shapiro,Onb-Whittaker functions, arXiv preprint[1806.00747]

work page arXiv
[22]

Shintani,On a Kronecker limit formula for real quadratic fields, Journal of the Faculty of Science, the University of Tokyo24(1977) 167–199

T. Shintani,On a Kronecker limit formula for real quadratic fields, Journal of the Faculty of Science, the University of Tokyo24(1977) 167–199

1977
[23]

V. P. Spiridonov,Essays on the theory of elliptic hypergeometric functions, Russian Mathematical Surveys63:3 (2008) 405–472,[0805.3135]. N. B.: Beijing Institute of Mathematical Sciences and Applications, Huairou district, Beijing, 101408, China G. S.: Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, 141980 Russia and Yerevan Physics Institu...

work page arXiv 2008