The Partial Theta Operator and Multivariate Generalized Lambert Series

TL;DR

This paper introduces a new Theta Partial operator and uses it to generalize Lambert series, including multivariate and bivariate forms, providing a broad framework for these series with elementary function examples.

Contribution

The paper develops the Theta Partial operator based on the λ-derivative and applies it to define various generalized Lambert series, extending existing series to multivariate and bivariate cases.

Findings

Defined the Theta Partial operator and generalized Lambert series.

Introduced multivariate and bivariate Lambert series.

Provided examples using elementary functions.

Abstract

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.