Binomial theorem and exponent for variables commuting as $yx=qxy$

TL;DR

This paper develops analogs of the binomial theorem and exponential function for variables that commute according to a q-commutation relation, extending classical algebraic identities to a quantum setting.

Contribution

It introduces new formulations of the binomial theorem and exponential function for q-commuting variables, expanding algebraic tools in quantum algebra.

Findings

Derived q-binomial theorem for non-commuting variables

Formulated exponential functions consistent with q-commutation

Extended classical identities to quantum algebra context

Abstract

We state analogs of the binomial theorem and the exponential function for variables $x$, $y$ commuting as $yx=qxy$.