标题： 偏Theta算子和多元广义Lambert级数 显示英文标题

标题： The Partial Theta Operator and Multivariate Generalized Lambert Series

摘要： 在本文中，我们引入了基于$\lambda$-导数算子$\mathbf{D}_{\lambda}$的 Theta 部分算子$\theta(y\mathbf{D}_{\lambda})$，并用它来定义以下 Lambert 级数\begin{equation*} \sum_{n=0}^{\infty}a_{n}\frac{x^{n+1}}{x-\lambda^ny}z^n. \end{equation*}的推广形式。此外，我们还定义了广义的 Lambert-Mehler 和 Lambert-Rogers 型级数、双重求和二元广义 Lambert 级数以及多元广义 Lambert 级数。通过使用初等函数提供了一组有趣的广义 Lambert 级数列表。 显示英文摘要

摘要： 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.