$q$-Rational and $q$-Real Binomial Coefficients

TL;DR

This paper extends $q$-binomial coefficients to $q$-rational and $q$-real numbers using continued fractions, establishing fundamental identities and exploring their algebraic properties.

Contribution

It introduces new $q$-binomial coefficients based on $q$-rational and $q$-real numbers and proves key identities like the $q$-Pascal and $q$-binomial theorem in this context.

Findings

Established $q$-Pascal identity for $q$-rational and $q$-real binomial coefficients

Derived $q$-binomial theorem in the new setting

Found additional identities including Chu--Vandermonde and $q$-Gamma function identities

Abstract

We consider $q$-binomial coefficients built from the $q$-rational and $q$-real numbers defined by Morier-Genoud and Ovsienko in terms of continued fractions. We establish versions of both the $q$-Pascal identity and the $q$-binomial theorem in this setting. These results are then used to find more identities satisfied by the $q$-analogues of Morier-Genoud and Ovsienko, including a Chu--Vandermonde identity and $q$-Gamma function identities.