Gaussian binomial coefficients with negative arguments

TL;DR

This paper extends the properties and interpretations of binomial coefficients to negative and Gaussian cases, including a generalized binomial theorem and Lucas' Theorem, revealing new algebraic and combinatorial insights.

Contribution

It generalizes binomial coefficient properties to negative and Gaussian cases, including a uniform binomial theorem and Lucas' Theorem extension.

Findings

Extension of binomial theorem to negative and Gaussian coefficients

Combinatorial interpretation for negative entries

Lucas' Theorem applies to Gaussian binomial coefficients

Abstract

Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas' Theorem on binomial coefficients modulo $p$ not only extends naturally to the case of negative entries, but even to the Gaussian case.