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arxiv: 2605.24010 · v1 · pith:ZGN7SZJRnew · submitted 2026-05-19 · 🧮 math.QA · math-ph· math.FA· math.MP

Topological analysis in mathcal{R}(p,q)-anisotropic sector and nuclear space on mathcal{R}(p,q)-quantum deformed algebra

Kaw\`eyim Lankpetre , Isiaka Aremua , Joseph D\'esir\'e Bukweli Kyemba This is my paper

Pith reviewed 2026-06-30 17:19 UTC · model grok-4.3

classification 🧮 math.QA math-phmath.FAmath.MP
keywords R(p,q)-deformationdeformed Gamma functionStirling asymptoticsentire functionsanisotropic sectorsnuclear spacesquantum algebrasCauchy-Hadamard theorem
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The pith

A Stirling-type asymptotic for the R(p,q)-deformed Gamma function produces quadratic exponential growth estimates that yield sharp bounds and classical inequality analogues for entire functions in the deformed setting.

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

The paper develops a topological and analytic framework for holomorphic functions under a general R(p,q)-deformation defined by a meromorphic kernel. It introduces the corresponding deformed Gamma function and establishes its Stirling-type asymptotic expansion, which is driven by the quadratic exponential growth of the deformed factorial. These estimates allow the construction of weighted Banach and Fréchet spaces with deformation-dependent pseudo-norms and the definition of anisotropic sectors. In this setting, the paper proves R(p,q)-versions of the Cauchy-Hadamard theorem, Borel-Carathéodory inequality, and Phragmén-Lindelöf principles. A reader would care because this provides tools for functional calculus in quantum-deformed algebras that may extend to problems in mathematical physics.

Core claim

The R(p,q)-deformed Gamma function admits a Stirling-type asymptotic expansion based on the asymptotics R!(p^n, q^n) ∼ exp(λ n²), which induces precise exponential quadratic growth estimates. These in turn yield sharp coefficient bounds and Cauchy-type inequalities for R(p,q)-entire functions, enabling the introduction of R(p,q)-weighted spaces, discs, and anisotropic sectors where analogues of classical complex analysis theorems hold.

What carries the argument

The R(p,q)-deformed Gamma function Γ_{R(p,q)} together with the associated weighted spaces and valuation maps, which translate the deformation geometry into growth controls for holomorphic functions.

Load-bearing premise

The meromorphic kernel must satisfy 0<q<p≤1 with R(1,1)=0 and R(p^n,q^n)>0 so that the deformed Gamma function is well-defined and positive.

What would settle it

Computing the deformed factorial for large n and checking whether its logarithm grows exactly quadratically as predicted, or verifying if a specific entire function violates the induced Cauchy inequality in the sector.

Figures

Figures reproduced from arXiv: 2605.24010 by Isiaka Aremua, Joseph D\'esir\'e Bukweli Kyemba, Kaw\`eyim Lankpetre.

Figure 1
Figure 1. Figure 1: R(p, q)−anisotropic sector SR of analyticity. Definition 4.3 (Fréchet R(p, q)-Algebra and nuclear space of functions) Let A be a commutative unital Fréchet algebra over C, with a countable family of seminorms {∥ · ∥m}m∈N defining its locally convex topology. Let N ⊂ O(C) be a nuclear space of R(p, q)-analytic functions, i.e. functions admitting R(p, q)-deformed power series expansions with convergence cont… view at source ↗
read the original abstract

The purpose of this article is to develop and analyze $\mathcal{R}(p,q)-$topological analysis of the classical nuclear space within the general framework of $\mathcal{R}(p,q)-$calculus. We begin by introducing the $\mathcal{R}(p,q)-$Gamma functions, establishing their main properties and their connection with the deformed factorials. We develop a rigorous analytic and functional-analytic framework for holomorphic functions governed by a general $\mathcal{R}(p,q)-$deformation, where $\mathcal{R}(u,v)$ is a meromorphic kernel satisfying $0<q<p\leq 1$, $\mathcal{R}(1,1)=0$, and $\mathcal{R}(p^n,q^n)>0$. A Stirling-type asymptotic expansion is established for the $\mathcal{R}(p,q)-$deformed Gamma function $\Gamma_{\mathcal{R}(p,q)}$, yielding precise exponential quadratic growth estimates driven by the asymptotics of the deformed factorial $\mathcal{R}!(p^n,q^n)\sim \exp(\lambda n^2)$. These asymptotics induce sharp coefficient bounds and Cauchy-type inequalities for $\mathcal{R}(p,q)-$entire functions. Based on these estimates, we introduce $\mathcal{R}(p,q)-$weighted Banach and Fr\'echet spaces of holomorphic functions, together with deformation dependent pseudo-norms and valuation maps. Within this setting, we define $\mathcal{R}(p,q)-$discs and anisotropic sectors adapted to the deformation geometry and prove $\mathcal{R}(p,q)-$analogues of the Cauchy-Hadamard theorem, the Borel-Carath\'eodory inequality and Phragm\'en-Lindel\"of type growth principles. These results contribute to the broader program of constructing a consistent functional calculus in $\mathcal{R}(p,q)-$quantum algebras, with potential applications to deformed fractional differential equations, operator theory, spectral problems, and non commutative models arising in mathematical physics.

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.

Referee Report

1 major / 0 minor

Summary. The paper develops an R(p,q)-deformed calculus framework starting from a meromorphic kernel R(u,v) satisfying 0<q<p≤1, R(1,1)=0 and R(p^n,q^n)>0. It defines an R(p,q)-Gamma function Γ_R(p,q) via the associated deformed factorial, claims a Stirling-type asymptotic expansion for this Gamma function driven by the quadratic-exponential growth R!(p^n,q^n)∼exp(λn²), derives sharp coefficient bounds and Cauchy inequalities for R(p,q)-entire functions, constructs R(p,q)-weighted Banach/Fréchet spaces with deformation-dependent pseudo-norms, introduces R(p,q)-discs and anisotropic sectors, and proves deformed analogues of the Cauchy-Hadamard theorem, Borel-Carathéodory inequality and Phragmén-Lindelöf growth principles.

Significance. If the key asymptotic claim holds under the stated hypotheses, the work supplies a coherent functional-analytic setting for holomorphic functions on R(p,q)-quantum algebras, including growth estimates and sectorial analysis that could support applications to deformed fractional DEs, operator theory and non-commutative models. The explicit construction of weighted spaces and valuation maps from the deformed factorial asymptotics would be a concrete contribution to the program of consistent functional calculus in deformed settings.

major comments (1)
  1. [the section establishing the Stirling-type asymptotic expansion for Γ_R(p,q) and the asymptotics of R!(p^n,q^n)] The central claim that the deformed factorial satisfies R!(p^n,q^n)∼exp(λn²) (and therefore yields a Stirling-type expansion for Γ_R(p,q)) is asserted for an arbitrary meromorphic kernel obeying only 0<q<p≤1, R(1,1)=0 and R(p^n,q^n)>0. These conditions alone do not force the partial sums of log R(p^k,q^k) to be asymptotically quadratic; generic meromorphic R can produce different growth rates. This gap is load-bearing for the coefficient bounds, Cauchy inequalities, weighted spaces, discs, sectors and all subsequent growth principles.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for pinpointing the foundational gap in the asymptotic justification. We address the concern directly below and will revise the manuscript to incorporate an explicit hypothesis that closes this gap while preserving the overall framework.

read point-by-point responses

  1. Referee: [the section establishing the Stirling-type asymptotic expansion for Γ_R(p,q) and the asymptotics of R!(p^n,q^n)] The central claim that the deformed factorial satisfies R!(p^n,q^n)∼exp(λn²) (and therefore yields a Stirling-type expansion for Γ_R(p,q)) is asserted for an arbitrary meromorphic kernel obeying only 0<q<p≤1, R(1,1)=0 and R(p^n,q^n)>0. These conditions alone do not force the partial sums of log R(p^k,q^k) to be asymptotically quadratic; generic meromorphic R can produce different growth rates. This gap is load-bearing for the coefficient bounds, Cauchy inequalities, weighted spaces, discs, sectors and all subsequent growth principles.

    Authors: We agree that the referee's observation is correct: the stated conditions on the meromorphic kernel R are insufficient by themselves to guarantee the quadratic growth ∑_{k=1}^n log R(p^k,q^k) ∼ λ n². In the submitted manuscript this asymptotic was introduced to derive the Stirling-type expansion and all downstream results, but without an additional hypothesis the claim does not hold for arbitrary kernels satisfying only the listed properties. In the revised version we will add an explicit standing hypothesis (H): there exists λ>0 such that log R!(p^n,q^n) = ∑_{k=1}^n log R(p^k,q^k) ∼ λ n² as n→∞. Under (H) the Stirling-type expansion for Γ_R(p,q) follows by standard summation arguments, and the coefficient bounds, Cauchy inequalities, weighted spaces, discs, sectors, and growth principles remain valid when conditioned on (H). We will also include concrete examples of kernels (e.g., suitable quadratic-exponential meromorphic functions) that satisfy both the original conditions and (H). This revision makes the logical structure rigorous without changing the paper's scope or conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations follow from definitions without reduction to inputs by construction

full rationale

The paper defines the R(p,q)-deformed Gamma function and factorials from the meromorphic kernel R(u,v) under the stated conditions (0<q<p≤1, R(1,1)=0, R(p^n,q^n)>0), then claims to establish Stirling-type asymptotics for Γ_R(p,q) driven by R!(p^n,q^n)∼exp(λn²) and derives subsequent bounds, inequalities, and spaces from those estimates. No quoted step shows a prediction equivalent to its inputs by construction, no fitted parameter renamed as prediction, and no load-bearing self-citation chain. The framework adapts standard analytic techniques to the deformation; the central claims are presented as derived rather than tautological. This is the most common honest finding for self-contained papers.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The paper relies on the definition of the R(p,q) kernel and the construction of the deformed factorial and Gamma function as foundational elements.

free parameters (1)
  • p,q
    Deformation parameters with 0<q<p≤1
axioms (1)
  • domain assumption R(u,v) is meromorphic kernel with R(1,1)=0 and R(p^n,q^n)>0
    Stated in the abstract as the general framework
invented entities (1)
  • R(p,q)-deformed Gamma function no independent evidence
    purpose: To generalize the Gamma function for the deformation
    Introduced in the paper

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Works this paper leans on

44 extracted references · 12 canonical work pages · 7 internal anchors

  1. [2]

    Jagannathan R and Srinivasa Rao K 2006 Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series arXiv:math/0602613

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  2. [3]

    Generalized Fractional Calculus and Applications; Longman—J

    Kiryakova, V. Generalized Fractional Calculus and Applications; Longman—J. Wiley, Harlow—New York, Chapman and Hall/CRC: London, UK, 1994

    1994

  3. [4]

    Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algo- rithmic Tables; Ellis Horwood: Chichester, UK, 1983

    Marichev, O.I. Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algo- rithmic Tables; Ellis Horwood: Chichester, UK, 1983

    1983

  4. [5]

    The H-Functions of One and Two Variables with Applications; South Asian Publs: New Delhi, India, 1982

    Srivastava, H.M.; Gupta, K.S.; Goyal, S.P. The H-Functions of One and Two Variables with Applications; South Asian Publs: New Delhi, India, 1982

    1982

  5. [6]

    Special Functions in Queuing Theory and Related Stochastic Pro- cesses; Academic Press: New York, NY, USA, 1982

    Srivastava, H.M.; Kashyap, B.R.K. Special Functions in Queuing Theory and Related Stochastic Pro- cesses; Academic Press: New York, NY, USA, 1982

    1982

  6. [7]

    Integrals and Series, Vol

    Prudnikov, A.P.; Brychkov, Y.; Marichev, O.I. Integrals and Series, Vol. 3: More Special Functions; Gordon and Breach Sci. Publ.: London, UK, 1992

    1992

  7. [8]

    The Hypergeometric Approach to Integral Transforms and Convolutions; Ser

    Yakubovich, S.; Luchko, Y. The Hypergeometric Approach to Integral Transforms and Convolutions; Ser. Mathematics and Its Applications 287; Kluwer Acad. Publ.: Dordrecht, The Netherlands; Boston, MA, USA; London, UK, 1994

    1994

  8. [9]

    Theory Appl., Vol

    Alok Jain, Altaf Ahmad Bhat, Renu Jain and Deepak Kumar Jain,(p,q)−Analogue of Mittag-Leffler Function with(p,q)−Laplace Transform, Anal. Theory Appl., Vol. 38, No. 3 (2022), pp. 351-360, DOI: 10.4208/ata.OA-2018-0014 September 2022

    work page doi:10.4208/ata.oa-2018-0014 2022

  9. [10]

    https://doi.org/10.3390/math9010106

    Virginia Kiryakova, A Guide to Special Functions in Fractional Calculus, Mathematics 2021, 9, 106. https://doi.org/10.3390/math9010106

    work page doi:10.3390/math9010106 2021

  10. [11]

    Orthogonal Polynomials and Special Functions; SIAM: Philadelphia, PA, USA, 1975

    Askey, R. Orthogonal Polynomials and Special Functions; SIAM: Philadelphia, PA, USA, 1975

    1975

  11. [12]

    (Ed.) Higher Transcendental Functions; McGraw Hill: New York, NY, USA, 1953–1955; Vols

    Erdélyi, A. (Ed.) Higher Transcendental Functions; McGraw Hill: New York, NY, USA, 1953–1955; Vols. 1, 2, 3

    1953

  12. [13]

    Handbook of Mathematical Functions with Formulas, Graphs, and Math- ematical Tables; National Bureau of Standards, 1964-Dover, 1965

    Abramowitz, M.; Stegun, A. Handbook of Mathematical Functions with Formulas, Graphs, and Math- ematical Tables; National Bureau of Standards, 1964-Dover, 1965

    1964

  13. [14]

    Hounkonnou M N and Bukweli Kyemba J D 2010 Generalized(R,p,q)−deformed Heisenberg algebras: coherent states and special functionsJ. Math. Phys.51063518 25

    2010

  14. [15]

    Hounkonnou M N, Howard F. A. and Kangni K. Particles andp−adic integrals ofSpin 1 2 : spin Lie group,R(ρ,q)−gamma andR(ρ,q)−beta functions, ghost and applications arXiv:2308.06363v1 [math.NT] 10 Aug 2023

    work page arXiv 2023

  15. [16]

    MelongFandWulkenhaarRGeneralizedHeisenberg-Virasoroalgebraandmatrixmodelsfromquantum algebra arXiv:1005.4309v1 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv

  16. [17]

    Bukweli Kyemba J D and Hounkonnou M N Characterization of(R,p,q)−deformed Rogers-Szegö polynomials: associated quantum algebras, deformed Hermite polynomials and relevant properties arXiv:1204.4705v1 [math-ph] 20 Apr 2012

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  17. [18]

    Hounkonnou M N and Ngompe Nkouankam E B 2007 New(p,q,µ,ν,f)−deformed statesJ. Phys. A: Math. Theor.4012113

    2007

  18. [19]

    Chari V and Pressley A 1994A guide to Quantum groups(Cambridge: Cambridge University Press)

  19. [20]

    N., Melong F

    Hounkonnou M. N., Melong F. and Melanija Mitrović.: Generalized Witt, Witt n algebras, Virasoro al- gebrasandKdVequations inducedfromR(p,q)−deformedquantumalgebras, e-printarXiv:2008.04778v1 [math-ph] 8 Aug 2020

    work page arXiv 2008

  20. [21]

    N., Melong F

    Hounkonnou M. N., Melong F. and Melanija Mitrović.: On a general method of constructing deformed quantum analogs of classical orthogonal polynomials, SUT Journal of mathematics (2020)

    2020

  21. [22]

    Hounkonnou M. N. and Melong F.:R(p,q)−deformed conformal Virasoro algebra,J. Maths. Phys.60, (2019)

    2019

  22. [23]

    Hounkonnou M. N. and Melong F.:R(p,q)−analogs of discrete distribution: general formalism and application, e-print arXiv:1901.01840v1 [math.PR](2019)

    work page arXiv 1901

  23. [24]

    Bukweli Kyemba and M.N

    J.D. Bukweli Kyemba and M.N. Hounkonnou, 2012(q; l,λ)-deformed Heisenberg algebra: coherent states, their statistics and geometry, Afr. Diaspora J. Math, 14 pp 38 -56

    2012

  24. [25]

    Hounkonnou M. N. and Melong F.:R(p,q)−deformed combinatorics: full characterization and illus- tration, arXiv:1906.03059v1 [math.GM](2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  25. [26]

    Hounkonnou M. N. and Kyemba Bukweli J. D.:R(p,q)-calculus: differentiation and integration,SUT. J. Math.Vol49n ◦2, 145-167 (2013)

    2013

  26. [27]

    On the $(p,q)$-Gamma and the $(p,q)$-Beta functions

    Sadjang P. N.: On the(p,q)−gamma and the(p,q)−beta functions. arXiv 2015, arXiv:1506.07394

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  27. [28]

    On the fundamental theorem of $(p,q)$-calculus and some $(p,q)$-Taylor formulas

    Sadjang P. N.: On the fundamental theorem of(p,q)−calculus and some(p,q)−Taylor formulas. arXiv 2013, arXiv:1309.3934v1

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  28. [29]

    Milovanovic G.V., Gupta V., Malik N.:(p,q)−Beta functions and applications in approximation. Bol. Soc. Mat. Mex. 2016

    2016

  29. [30]

    Some completely monotonic properties for the $(p,q )$-gamma function

    V. Krasniqi, F. Merovci.: Some completely monotonic properties for the(p,q)−Gamma function, http://arxiv.org/abs/1407.4231v1, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  30. [31]

    Kiryakova, V.Generalized Fractional Calculus and Applications, Longman, 1994

    1994

  31. [32]

    and others.Fractional Calculus Operators and Applications to Differential Equations, 1991

    Saigo, M. and others.Fractional Calculus Operators and Applications to Differential Equations, 1991

    1991

  32. [33]

    et al.(p,q)-Calculus and Quantum Calculus Extensions, SIGMA, 2023

    Chakrabarty, A. et al.(p,q)-Calculus and Quantum Calculus Extensions, SIGMA, 2023

    2023

  33. [34]

    F. H. Jackson,Onq-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh46(1908), 253–281

    1908

  34. [35]

    Gasper and M

    G. Gasper and M. Rahman,Basic Hypergeometric Series, Cambridge University Press, Cambridge, 2004. 26

    2004

  35. [36]

    Jagannathan and K

    R. Jagannathan and K. Srinivasa Rao,Two-parameter quantum algebras, twin-basic numbers, and associated special functions, Proc. Int. Conf. on Number Theory and Mathematical Physics, 2005

    2005

  36. [37]

    Chakrabarti and R

    R. Chakrabarti and R. Jagannathan,A(p,q)-oscillator realization of two-parameter quantum algebras, J. Phys. A24(1991), L711–L718

    1991

  37. [38]

    Burban and A

    I. Burban and A. Klimyk,Two-parameter quantum algebras and deformed oscillators, J. Phys. A26 (1993), 4871–4882

    1993

  38. [39]

    M. N. Hounkonnou and J. B. Ngompe,R(p,q)-deformed algebras and special functions, J. Math. Phys. 50(2009), 123502

    2009

  39. [40]

    Exton,q-Hypergeometric Functions and Applications, Ellis Horwood, Chichester, 1983

    H. Exton,q-Hypergeometric Functions and Applications, Ellis Horwood, Chichester, 1983

    1983

  40. [41]

    T. H. Koornwinder,q-special functions and quantum groups, Contemp. Math.134(1992), 73–94

    1992

  41. [42]

    V. G. Drinfeld,Quantum groups, Proc. Int. Congr. Math. (Berkeley, 1986), 798–820

    1986

  42. [43]

    Jimbo,Aq-difference analogue ofU(g)and the Yang–Baxter equation, Lett

    M. Jimbo,Aq-difference analogue ofU(g)and the Yang–Baxter equation, Lett. Math. Phys.10(1985), 63–69

    1985

  43. [44]

    Herrmann,Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore, 2014

    R. Herrmann,Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore, 2014

    2014

  44. [45]

    Podlubny,Fractional Differential Equations, Academic Press, San Diego, 1999

    I. Podlubny,Fractional Differential Equations, Academic Press, San Diego, 1999. 27

    1999