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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.24010 |
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Table of Contents:
- 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 $Γ_{\mathcal{R}(p,q)}$, yielding precise exponential quadratic growth estimates driven by the asymptotics of the deformed factorial $\mathcal{R}!(p^n,q^n)\sim \exp(λn^2)$. 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 Fréchet 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-Carathéodory inequality and Phragmén-Lindelöf 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.