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The factorial function and generalizations, extended

Jeffrey C. Lagarias, Wijit Yangjit

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.

The factorial function and generalizations, extended

Abstract

This paper presents an extension of Bhargava's theory of factorials associated to any nonempty subset of . Bhargava's factorials are invariants, constructed using the notion of -orderings of where is a prime. This paper defines -orderings of any nonempty subset of for all integers , as well as "extreme" cases and . It defines generalized factorials and generalized binomial coefficients as nonnegative integers, for all nonempty and allowing only in . It computes -ordering invariants when is and when is the set of all primes.
Paper Structure (41 sections, 34 theorems, 138 equations, 4 tables)

This paper contains 41 sections, 34 theorems, 138 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Bhargava's $p$-orderings and generalized factorials
  3. Bhargava's $p$-orderings of nonempty subsets $S$ of $\mathbb{Z}$
  4. Bhargava's generalized factorials and generalized binomial coefficients associated to nonempty subsets $S$ of $\mathbb{Z}$
  5. Main results
  6. $b$-orderings and $b$-exponent sequences for arbitrary bases $b \ge 2$ in $\mathbb{Z}$.
  7. Max-min criterion for $b$-exponent sequence invariants and majorization
  8. Generalized factorials from $b$-exponent sequences in $\mathbb{Z}$.
  9. Generalized-integers and generalized binomial coefficients
  10. Contents of the paper
  11. ${t}$-orderings and ${t}$-exponent sequences in the power series ring $\mathbb{Q}[[t]]$
  12. ${t}$-orderings of an arbitrary subset of $\mathbb{Q}[[t]]$
  13. Well-definedness of $t$-exponent sequences of arbitrary nonempty $U\subseteq \mathbb{Q}[[t]]$; max-min-characterization
  14. Proof of max-min-characterization for $t$-exponent sequences of nonempty $U\subseteq\mathbb{Q}[[t]]$
  15. Properties of the associated ${t}$-exponent sequences
  16. ...and 26 more sections

Key Result

Theorem 2.2

For a nonempty subset $S$ of $\mathbb{Z}$ and a prime $p$, the associated $p$-sequence of a $p$-ordering of $S$ is independent of the choice of $p$-ordering.

Theorems & Definitions (74)

  • Definition 2.1
  • Theorem 2.2: Bhargava Bhar:00
  • Theorem 2.3: Bhargava Bhar:00
  • Theorem 2.4: Bhargava Bhar:00
  • Theorem 2.5: Bhargava Bhar:00
  • Theorem 2.6: Bhargava Bhar:00
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Well-definedness of the $b$-exponent sequence of $S$ in $\mathbb{Z}$
  • Example 3.4
  • ...and 64 more