Lucas recurrences and Lambert series in coprime double sums
Abstract
For the Lucas numbers L0=2, L1=1, and Ln=Ln−1+Ln−2, we obtain ∑substackn,m≥1 (n,m)=11/(L2nL2mL2(n+m))=1/45. This follows from the absolutely convergent identity ∑substackn,m≥1 (n,m)=1(qn+m)/((1+qn)(1+qm)(1+qn+m)) =q2/((1−q2)2), |q|<1. We give a self-contained proof by local cancellation on the Calkin--Wilf tree, as a specialization of Hata's Farey-interval telescoping principle. Specializations yield a hyperbolic-cosine identity ∑substackn,m≥1 (n,m)=1 1/(cosh(nx)cosh(mx)cosh((n+m)x)) =2/sinh2(2x), x>0 and weighted sums for canonical second-order Lucas recurrences. Decomposing arbitrary positive pairs according to their greatest common divisor gives an unrestricted sum expressed by a Lambert series and the quasimodular Eisenstein series E2. Conversely, Möbius inversion recovers the primitive identity from a double Lambert-series result of Cui, Kumar, and Singh.
Related
- Infinite Series About Reciprocal Products of Fibonacci and Lucas Numbers
- Estimates of Lucas series and expansions in non-integer bases
- Initial Runs in Integer Compositions: Lambert Series and Generalized Divisor Sums
- Inverse trigonometric and hyperbolic identities,series and products for Lucas sequences
- Double shuffle and naive duality for multiple Dirichlet series
BibTeX
@misc{kalinin2026lucas,
title = {{Lucas recurrences and Lambert series in coprime double sums}},
author = {Nikita Kalinin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26949},
eprint = {2610.26949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26949},
}
RIS
TY - UNPB
TI - Lucas recurrences and Lambert series in coprime double sums
AU - Nikita Kalinin
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.26949
UR - https://arxiv.org/abs/2610.26949
AB - For the Lucas numbers L_0=2, L_1=1, and L_n=L_n−1+L_n−2, we obtain ∑_substackn,m≥1 (n,m)=11/(L_2nL_2mL_2(n+m))=1/45. This follows from the absolutely convergent identity ∑_substackn,m≥1 (n,m)=1(q^n+m)/((1+q^n)(1+q^m)(1+q^n+m)) =q^2/((1−q^2)^2), |q|<1. We give a self-contained proof by local cancellation on the Calkin--Wilf tree, as a specialization of Hata's Farey-interval telescoping principle. Specializations yield a hyperbolic-cosine identity ∑_substackn,m≥1 (n,m)=1 1/(cosh(nx)cosh(mx)cosh((n+m)x)) =2/sinh^2(2x), x>0 and weighted sums for canonical second-order Lucas recurrences. Decomposing arbitrary positive pairs according to their greatest common divisor gives an unrestricted sum expressed by a Lambert series and the quasimodular Eisenstein series E_2. Conversely, Möbius inversion recovers the primitive identity from a double Lambert-series result of Cui, Kumar, and Singh.
ER -