A Resolvent Approach to Generalized Lambert Series and $q$-Series Identities

We introduce a partial-theta-type \(q\)-operator $$ \Theta(yD_q)=\sum_{n\ge0}q^{\binom n2}y^nD_q^n, \qquad D_qf(x)=\frac{f(qx)}{x}, $$ and show that it admits the resolvent representation $$ \Theta(yD_q)=\left(I-\frac yxM_q\right)^{-1}, $$ where $M_qf(x)=f(qx)$.

This identity provides a unified operational framework for generalized Lambert series and their Mehler, Rogers, and bilateral analogues.

Starting from ordinary and bilateral generating functions, we obtain Lambert-type expansions and derive consequences involving basic hypergeometric series, Ramanujan's ${}_1\psi_1$ summation, and Kronecker-type theta identities.

The method gives a compact way to generate families of $q$-series identities from a first-order $q$-difference resolvent.