The solutions of four $q$-functional equations

TL;DR

This paper derives solutions for four specific $q$-functional equations and expresses them using $q$-operator forms, providing conditions for applying $q$-operator methods effectively.

Contribution

It introduces solutions to four $q$-functional equations and establishes conditions for their use with $q$-operator techniques.

Findings

Solutions expressed in $q$-operator forms

Provides sufficient conditions for $q$-operator methods

Enhances understanding of $q$-functional equations

Abstract

In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.