A Matrix-Theoretic Representation and Integrality Properties of General Franel Sums
Abstract
We investigate the integrality and structural properties of combinatorial sequences associated with the general Franel numbers Fl(m) = ∑k=0l binomlkm for exponents m ≥ 3. To avoid the emergence of fractional coefficients common in classical recurrence verification, we introduce an algebraic matrix framework operating entirely over the ring of integers ℤ. By constructing a unipotent transition matrix within a graded structure, we show that the intermediate variations satisfy specific boundary conditions under the matrix commutator. We prove that the underlying linear Diophantine system is strictly lower-triangular with unit diagonal entries, which ensures that the resulting coefficients are integers obtained via division-free back-substitution. This structural framework provides a matrix-theoretic formulation of the underlying divisibility properties, showing that the terminal component of the transformed vector reflects the global scalar invariant factorization (TcanPcomb(l))l = l3+mΛl, where Λl ∈ ℤ. Explicit numerical examples for small dimensions are provided to illustrate the exactness of these integer relations over ℤ.
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BibTeX
@misc{aitelhad2026matrixtheoretic,
title = {{A Matrix-Theoretic Representation and Integrality Properties of General Franel Sums}},
author = {Abdellatif Aitelhad},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.13140/rg.2.2.22039.53921/1},
url = {https://hal.science/hal-05635299v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Matrix-Theoretic Representation and Integrality Properties of General Franel Sums
AU - Abdellatif Aitelhad
PY - 2026
DA - 2026/10/23
PB - HAL
DO - 10.13140/rg.2.2.22039.53921/1
UR - https://hal.science/hal-05635299v2
AB - We investigate the integrality and structural properties of combinatorial sequences associated with the general Franel numbers F_l^(m) = ∑_k=0^l binomlk^m for exponents m ≥ 3. To avoid the emergence of fractional coefficients common in classical recurrence verification, we introduce an algebraic matrix framework operating entirely over the ring of integers ℤ. By constructing a unipotent transition matrix within a graded structure, we show that the intermediate variations satisfy specific boundary conditions under the matrix commutator. We prove that the underlying linear Diophantine system is strictly lower-triangular with unit diagonal entries, which ensures that the resulting coefficients are integers obtained via division-free back-substitution. This structural framework provides a matrix-theoretic formulation of the underlying divisibility properties, showing that the terminal component of the transformed vector reflects the global scalar invariant factorization (T_canP_comb(l))_l = l^3+mΛ_l, where Λ_l ∈ ℤ. Explicit numerical examples for small dimensions are provided to illustrate the exactness of these integer relations over ℤ.
ER -