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

Abstract

We introduce a partial-theta-type \(q\)-operator $$\Theta (y{D}_q)=\sum _{n\ge 0}{q}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}{y}^n{D}_q^n,D_{q}f(x)=\frac{f(qx)}{x},$$ and show that it admits the resolvent representation $$\Theta (y{D}_q)={\left(I-\frac{y}{x}{M}_q\right)}^{-1},$$ where $M_{q}f(x)=f(qx)$. 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.