The power quantum calculus and variational problems

TL;DR

This paper introduces a new power difference calculus based on a novel operator, explores its properties, and applies it to power quantum Lagrangian systems to derive related Euler–Lagrange equations.

Contribution

It presents a new power difference operator and integral, and applies these to formulate and analyze power quantum Lagrangian systems.

Findings

Defined the operator $D_{n,q}$ and its inverse integral.

Proved properties of the new operator and integral.

Derived $n,q$-Euler–Lagrange equations for power quantum systems.

Abstract

We introduce the power difference calculus based on the operator $D_{n,q} f(t) = \frac{f(qt^n)-f(t)}{qt^n -t}$, where $n$ is an odd positive integer and $0<q<1$. Properties of the new operator and its inverse --- the $d_{n,q}$ integral --- are proved. As an application, we consider power quantum Lagrangian systems and corresponding $n,q$-Euler--Lagrange equations.