Towards characterising polynomiality of $\frac{1-q^b}{1-q^a}{n\brack m}$ and applications

TL;DR

This paper characterizes when certain q-binomial coefficients are polynomials with integer coefficients, providing new divisibility results and unifying existing literature on polynomiality conditions.

Contribution

It offers a complete characterization for the polynomiality of specific q-binomial coefficients and introduces new divisibility properties, unifying previous results.

Findings

Characterization of polynomiality conditions for q-binomial coefficients

New divisibility properties for binomial coefficients

A novel divisibility result involving roots of unity

Abstract

In this note we shall give conditions which guarantee that $\frac{1-q^b}{1-q^a}{n\brack m}\in\mathbb{Z}[q]$ holds. We shall provide a full characterisation for $\frac{1-q^b}{1-q^a}{ka\brack m}\in\mathbb{Z}[q]$. This unifies a variety of results already known in literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity.