The authors prove an identity generalizing the Amdeberhan-Andrews-Ballantine conjecture on double Lambert series using coefficients from the generalized divisor function σ_k(n).
Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for $E_2(q)$
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series (\textit{J. Combin. Theory Series A} \textbf{221} (2026), Paper No. 106154). More precisely, they conjectured that \[ [q^{N2^a}] \sum_{m,k\geq 1} \frac{q^{mk2^a}}{(1+q^{k2^{a-1}})(1-q^{2m-1})} =\sigma_1(N), \] where $\sigma_1(N)$ is the sum of all the positive divisors of $N$. We provide two proofs of this conjecture. One of the approach leads us to derive a new representation of quasi-modular forms $E_2(q)$.
fields
math.NT 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Establishes transformation formulae for double Lambert series with applications to conjectures by Andrews, Dixit, Schultz, Yee and Amdeberhan et al.
citing papers explorer
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A Generalization of the Amdeberhan-Andrews-Ballantine Conjecture
The authors prove an identity generalizing the Amdeberhan-Andrews-Ballantine conjecture on double Lambert series using coefficients from the generalized divisor function σ_k(n).
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Transformation Formulae and Applications for Double Lambert Series
Establishes transformation formulae for double Lambert series with applications to conjectures by Andrews, Dixit, Schultz, Yee and Amdeberhan et al.