On the least common multiple of $q$-binomial coefficients

TL;DR

This paper proves a new identity relating the least common multiple of all $q$-binomial coefficients for a given n to the least common multiple of $q$-integers, extending a known classical identity to the $q$-analogue setting.

Contribution

It establishes a $q$-analogue of Farhi's classical identity, connecting the least common multiples of $q$-binomial coefficients and $q$-integers.

Findings

Proves the identity for the least common multiple of $q$-binomial coefficients.

Extends classical number theory identities to the $q$-analogue context.

Provides a new perspective on the structure of $q$-binomial coefficients.

Abstract

In this paper, we prove the following identity $$ \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, $$ where ${n\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].