On the use of Mellin transform to a class of q-difference-differential equations

TL;DR

This paper investigates the application of Mellin transforms to solve a specific class of q-difference-differential equations, demonstrating that Mellin transforms of q-derivatives resemble classical derivatives, which could facilitate solving these equations.

Contribution

It derives the Mellin transform of q-derivatives and shows its similarity to classical derivatives, opening new avenues for solving q-difference equations.

Findings

Mellin transform of q-derivatives closely resembles classical derivatives

The Mellin transform can be effectively used to solve certain q-difference equations

The results bridge classical analysis and q-calculus methods

Abstract

We explore the possibility of using the method of classical integral transforms to solve a class of $q$-difference-differential equations. The Laplace and the Mellin transform of $q$-derivatives are derived. The results show that the Mellin transform of the $q$-derivative resembles most closely the corresponding expression in classical analysis, and it could therefore be useful in solving certain $q$-difference equations.