The U-Derivative Associated with Linear Recurrence Sequences
Abstract
We associate a canonical operator to every linear recurrence sequence admitting a distinct-root Binet representation Un=∑j=1kAjλjn. The construction is based on the spectral Binet operator ℬU=∑j=1kAjTλj, where Tλjf(x)=f(λjx) is the dilation operator associated with the characteristic root λj. The corresponding recurrence derivative is defined by DU=x−1ℬU. Its fundamental property is the monomial identity DU(xn)=Unxn−1, which establishes a direct correspondence between recurrence sequences and degree-lowering operators. We develop the operator theory of DU, including iteration, a spectral product rule, the associated Leibniz defect, and a quotient rule. The construction reveals a natural bifurcation determined by the value U0, producing the Ward and non-Ward regimes as intrinsic operatorial phenomena. The product structure is shown to be governed by k−2 residual spectral channels, leading to notions of spectral cancellation and spectral balance. As an application, we obtain a complete qualitative classification of second-order recurrence calculi. Using Horadam sequences as a universal second-order model, we prove that every such calculus is determined by the invariant triple (W0,Δ, ZW), where W0 identifies the Ward/non-Ward type, Δ determines the spectral geometry, and ZW determines regularity or singularity.
Related
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BibTeX
@misc{lopez2026uderivative,
title = {{The $\mathrmU$-Derivative Associated with Linear Recurrence Sequences}},
author = {Ronald Orozco López},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20593},
eprint = {2610.20593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20593},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The U-Derivative Associated with Linear Recurrence Sequences
AU - Ronald Orozco López
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20593
UR - https://arxiv.org/abs/2610.20593
AB - We associate a canonical operator to every linear recurrence sequence admitting a distinct-root Binet representation U_n=∑_j=1^kA_jλ_j^n. The construction is based on the spectral Binet operator ℬ_U=∑_j=1^kA_jT_λ_j, where T_λ_jf(x)=f(λ_jx) is the dilation operator associated with the characteristic root λ_j. The corresponding recurrence derivative is defined by D_U=x^−1ℬ_U. Its fundamental property is the monomial identity D_U(x^n)=U_nx^n−1, which establishes a direct correspondence between recurrence sequences and degree-lowering operators. We develop the operator theory of D_U, including iteration, a spectral product rule, the associated Leibniz defect, and a quotient rule. The construction reveals a natural bifurcation determined by the value U_0, producing the Ward and non-Ward regimes as intrinsic operatorial phenomena. The product structure is shown to be governed by k−2 residual spectral channels, leading to notions of spectral cancellation and spectral balance. As an application, we obtain a complete qualitative classification of second-order recurrence calculi. Using Horadam sequences as a universal second-order model, we prove that every such calculus is determined by the invariant triple (W_0,Δ, Z_W), where W_0 identifies the Ward/non-Ward type, Δ determines the spectral geometry, and Z_W determines regularity or singularity.
ER -