The Partial Theta Operator and Multivariate Generalized Lambert Series

In this paper, we introduce the Theta Partial operator $\theta(y\mathbf{D}_{\lambda})$, based on the $\lambda$-derivative operator $\mathbf{D}_{\lambda}$, and use it to define the following generalization of the Lambert series \begin{equation*} \sum_{n=0}^{\infty}a_{n}\frac{x^{n+1}}{x-\lambda^ny}z^n. \end{equation*} Also, we define generalized Lambert-Mehler and Lambert-Rogers type series, double-sum bivariate generalized Lambert series and multivariate generalized Lambert series. A list of interesting generalized Lambert series is provided by using elementary functions.