The Partial Theta Operator and Multivariate Generalized Lambert Series
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912493276758016 |
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| author | López, Ronald Orozco |
| author_facet | López, Ronald Orozco |
| contents | In this paper, we introduce the Theta Partial operator $θ(y\mathbf{D}_λ)$, based on the $λ$-derivative operator $\mathbf{D}_λ$, 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-λ^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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15160 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Partial Theta Operator and Multivariate Generalized Lambert Series López, Ronald Orozco Number Theory 33E20, 05A15, 11B39, 11B65 In this paper, we introduce the Theta Partial operator $θ(y\mathbf{D}_λ)$, based on the $λ$-derivative operator $\mathbf{D}_λ$, 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-λ^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. |
| title | The Partial Theta Operator and Multivariate Generalized Lambert Series |
| topic | Number Theory 33E20, 05A15, 11B39, 11B65 |
| url | https://arxiv.org/abs/2507.15160 |