An Algebraic Formalization of Franel Modules: Graded Filtrations and Fully Faithful Functors over Z
Abstract
We construct a rigorous algebraic framework for the formalization of unipotent matrix transformations associated with the general Franel numbers Fl(m) = Pl k=0 klm for exponents m ≥ 3. Instead of introducing continuous differential geometries, we interpret the matrix recurrences through the structure of graded module filtrations and exact linear sequences over the ring of integers Z. We prove that the unique integrality and existence of the unique gauge vector a ∈ Zl derived in discrete computation settings is a direct structural consequence of the algebraic vanishing of higher-order extension modules over principal ideal domains. Furthermore, we establish a covariant linear functor F mapping the category of holonomic hypergeometric modules onto the category of filtered matricial modules, proving that this functor is strictly fully faithful over Z. Finally, this functorial formulation is applied to two algebraic specializations: the Legendre transform mapping, which regularizes the higher-order recurrence expansions of the historical Ap´ery numbers, and the non-commutative polynomial extension over the ring Z[q], where the structural gauge components establish a stable integer polynomial containment ak(q) ∈ Z[q]..
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BibTeX
@misc{aitelhad2026algebraic,
title = {{An Algebraic Formalization of Franel Modules: Graded Filtrations and Fully Faithful Functors over Z}},
author = {Abdellatif Aitelhad},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.13140/rg.2.2.19229.88809},
url = {https://hal.science/hal-05638285v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Algebraic Formalization of Franel Modules: Graded Filtrations and Fully Faithful Functors over Z
AU - Abdellatif Aitelhad
PY - 2026
DA - 2026/10/15
PB - HAL
DO - 10.13140/rg.2.2.19229.88809
UR - https://hal.science/hal-05638285v2
AB - We construct a rigorous algebraic framework for the formalization of unipotent matrix transformations associated with the general Franel numbers Fl(m) = Pl k=0 klm for exponents m ≥ 3. Instead of introducing continuous differential geometries, we interpret the matrix recurrences through the structure of graded module filtrations and exact linear sequences over the ring of integers Z. We prove that the unique integrality and existence of the unique gauge vector a ∈ Zl derived in discrete computation settings is a direct structural consequence of the algebraic vanishing of higher-order extension modules over principal ideal domains. Furthermore, we establish a covariant linear functor F mapping the category of holonomic hypergeometric modules onto the category of filtered matricial modules, proving that this functor is strictly fully faithful over Z. Finally, this functorial formulation is applied to two algebraic specializations: the Legendre transform mapping, which regularizes the higher-order recurrence expansions of the historical Ap´ery numbers, and the non-commutative polynomial extension over the ring Z[q], where the structural gauge components establish a stable integer polynomial containment ak(q) ∈ Z[q]..
ER -