On the $q$-factorization of power series
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916903669202944 |
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| author | Schneider, Robert Sills, Andrew V. Waldron, Hunter |
| author_facet | Schneider, Robert Sills, Andrew V. Waldron, Hunter |
| contents | Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note $q$-analogues of our enumeration formulas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $q$-factorization of power series Schneider, Robert Sills, Andrew V. Waldron, Hunter Combinatorics Number Theory 30B10, 11P81, 40A20 Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note $q$-analogues of our enumeration formulas. |
| title | On the $q$-factorization of power series |
| topic | Combinatorics Number Theory 30B10, 11P81, 40A20 |
| url | https://arxiv.org/abs/2501.18744 |