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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.18776 |
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| _version_ | 1866912069651005440 |
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| author | Petkova, Ioana |
| author_facet | Petkova, Ioana |
| contents | In this paper the operator $A = u(z)\frac{d}{dz}$ is considered, where $u$ is an entire or meromorphic function in the complex plane. The expansion of $A^{k}$ ($k\geq1$) with the help of the powers of the differential operator $D=\frac{d}{dz}$ is obtained, and it is shown that this expansion depends on special numbers. Connections between these numbers and known combinatorial numbers are given. Some special cases of the operator $A$, corresponding to $u(z) = z$, $u(z) = e^{z}$, $u(z) = \frac{1}{z}$, are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18776 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers Petkova, Ioana Combinatorics In this paper the operator $A = u(z)\frac{d}{dz}$ is considered, where $u$ is an entire or meromorphic function in the complex plane. The expansion of $A^{k}$ ($k\geq1$) with the help of the powers of the differential operator $D=\frac{d}{dz}$ is obtained, and it is shown that this expansion depends on special numbers. Connections between these numbers and known combinatorial numbers are given. Some special cases of the operator $A$, corresponding to $u(z) = z$, $u(z) = e^{z}$, $u(z) = \frac{1}{z}$, are considered. |
| title | Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.18776 |