Discrete q-derivatives and symmetries of q-difference equations

TL;DR

This paper extends umbral calculus to q-difference equations, enabling solutions and symmetry analysis for these equations, including the q-heat equation and second order q-difference equations.

Contribution

The paper introduces a q-umbral calculus framework that generalizes shift invariant difference operators to q-difference operators, facilitating solutions and symmetry analysis.

Findings

Developed a q-umbral calculus framework

Constructed symmetry solutions for the q-heat equation

Solved a linear second order q-difference equation

Abstract

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation and to solve a linear ordinary second order q-difference equation