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arXiv · 2310.12949

The factorial function and generalizations, extended

Jeffrey C. Lagarias·Wijit Yangjit
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Abstract

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.

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BibTeXRIS

Jeffrey C. Lagarias, Wijit Yangjit. 2023-10-19. The factorial function and generalizations, extended. https://doi.org/10.1016/j.jnt.2023.12.004

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