Generalized Lambert Series Identities and Applications in Rank Differences

In this article, we prove two identities of generalized Lambert series. By introducing what we call $\mathcal{S}$-series, we establish relationships between multiple generalized Lambert series and multiple infinite products. Compared with Chan's work, these new identities are useful in generating various formulas for generalized Lambert series with the same poles. Using these formulas, we study the 3-dissection properties of ranks for overpartitions modulo 6. In this case, $-1$ appears as a unit root, so that double poles occur. We also relate these ranks to the third order mock theta functions $\omega(q)$ and $\rho(q)$