Two Proofs of a Conjecture of Amdeberhan, Andrews and Ballantine for double Lambert series and a new Representation for $E_2(q)$

In this note, we prove a recent conjecture of Amdeberhan, Andrews and Ballantine concerning a double Lambert series ($\mathit{\text{J.}}\mathit{\text{Combin.}}\mathit{\text{Theory}}\mathit{\text{Series}}\mathit{\text{A}}$ $\mathbf{\text{221}}$ (2026), Paper No. 106154). More precisely, they conjectured that $$[q^{N{2}^a}]\sum _{m,k\ge 1}\frac{q^{mk{2}^a}}{(1+q^{k{2}^{a-1}})(1-q^{2m-1})}={\sigma }_1(N),$$ 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 $E_2(q)$.