  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 $\rho(q)$
