Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2023
- DOI:
- 10.48550/arXiv.2311.18776
- arXiv:
- arXiv:2311.18776
- Bibcode:
- 2023arXiv231118776P
- Keywords:
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- Mathematics - Combinatorics