Old and new powerful tools for the normal ordering problem and noncommutative binomials
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916433035788288 |
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| author | Beauduin, Kei |
| author_facet | Beauduin, Kei |
| contents | In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts for the exponential identity $e^{A+B} = e^A e^B$ and the binomial theorem $(A+B)^n = \sum \binom{n}{k} A^k B^{n-k}$ when the commutator $[B, A]$ is either an arbitrary quadratic polynomial or a monomial in $A$ or $B$. Analogous formulas are found when the commutator is bivariate. Furthermore, we introduce a novel operator bridging between the normal and antinormal ordered forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03001 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Old and new powerful tools for the normal ordering problem and noncommutative binomials Beauduin, Kei Combinatorics Quantum Algebra 05A10, 05A19, 11B73, 11B75, 11B65, 13N15, 16S32, 34A05, 34A12 In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts for the exponential identity $e^{A+B} = e^A e^B$ and the binomial theorem $(A+B)^n = \sum \binom{n}{k} A^k B^{n-k}$ when the commutator $[B, A]$ is either an arbitrary quadratic polynomial or a monomial in $A$ or $B$. Analogous formulas are found when the commutator is bivariate. Furthermore, we introduce a novel operator bridging between the normal and antinormal ordered forms. |
| title | Old and new powerful tools for the normal ordering problem and noncommutative binomials |
| topic | Combinatorics Quantum Algebra 05A10, 05A19, 11B73, 11B75, 11B65, 13N15, 16S32, 34A05, 34A12 |
| url | https://arxiv.org/abs/2405.03001 |