The factorial function and generalizations, extended
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917603385016320 |
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| author | Lagarias, Jeffrey C. Yangjit, Wijit |
| author_facet | Lagarias, Jeffrey C. Yangjit, Wijit |
| contents | This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset $S$ of $\mathbb{Z}$. Bhargava's factorials $k!_S$ are invariants, constructed using the notion of $p$-orderings of $S$ where $p$ is a prime. This paper defines $b$-orderings of any nonempty subset $S$ of $\mathbb{Z}$ for all integers $b\ge2$, as well as "extreme" cases $b=1$ and $b=0$. It defines generalized factorials $k !_{S,T}$ and generalized binomial coefficients $\binom{k+\ell}{k}_{S,T}$ as nonnegative integers, for all nonempty $S$ and allowing only $b$ in $T\subseteq\mathbb{N}$. It computes $b$-ordering invariants when $S$ is $\mathbb{Z}$ and when $S$ is the set of all primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12949 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The factorial function and generalizations, extended Lagarias, Jeffrey C. Yangjit, Wijit Number Theory 11B65, 11N80, 13F25 This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset $S$ of $\mathbb{Z}$. Bhargava's factorials $k!_S$ are invariants, constructed using the notion of $p$-orderings of $S$ where $p$ is a prime. This paper defines $b$-orderings of any nonempty subset $S$ of $\mathbb{Z}$ for all integers $b\ge2$, as well as "extreme" cases $b=1$ and $b=0$. It defines generalized factorials $k !_{S,T}$ and generalized binomial coefficients $\binom{k+\ell}{k}_{S,T}$ as nonnegative integers, for all nonempty $S$ and allowing only $b$ in $T\subseteq\mathbb{N}$. It computes $b$-ordering invariants when $S$ is $\mathbb{Z}$ and when $S$ is the set of all primes. |
| title | The factorial function and generalizations, extended |
| topic | Number Theory 11B65, 11N80, 13F25 |
| url | https://arxiv.org/abs/2310.12949 |