A symmetric quantum calculus

TL;DR

This paper develops a symmetric quantum calculus by introducing the $oldsymbol{ extalpha,eta}$-symmetric difference derivative and N"orlund sum, generalizing forward and backward $h$-calculus for broader mathematical applications.

Contribution

It introduces a new symmetric quantum calculus framework that unifies and extends existing difference calculus methods.

Findings

Defines the $oldsymbol{ extalpha,eta}$-symmetric difference derivative.

Establishes the $oldsymbol{ extalpha,eta}$-symmetric N"orlund sum.

Shows the calculus generalizes forward and backward $h$-calculus.

Abstract

We introduce the $\alpha,\beta$-symmetric difference derivative and the $\alpha,\beta$-symmetric N\"orlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward $h$-calculus.