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.
Categories: math.NT