Generalized Lambert Series Identities and Applications in Rank Differences

TL;DR

This paper proves new identities for generalized Lambert series, introduces $ ext{S}$-series to connect series and products, and applies these results to overpartition ranks and mock theta functions.

Contribution

It introduces $ ext{S}$-series to relate Lambert series and infinite products, enabling new formulas for rank differences in overpartitions.

Findings

Derived formulas for generalized Lambert series with common poles

Analyzed 3-dissection properties of overpartition ranks modulo 6

Connected rank functions to third order mock theta functions

Abstract

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 $\rho(q)$