D-polynomials and Taylor formula in quantum calculus

TL;DR

This paper introduces a generalized quantum calculus framework using a new divided difference operator, establishing polynomial concepts and a Taylor expansion within this algebraic setting.

Contribution

It proposes a novel quantum calculus based on the operator $D_{\sigma}^{ au}$, extending existing $h$- and $q$-calculus with new polynomial and Taylor formula concepts.

Findings

Defined $D_{\sigma}^{ au}$-polynomials

Proved a Taylor formula in this calculus

Generalized quantum derivative operator

Abstract

Quantum calculus based on the right invertible divided difference operator $D_{\sigma}^{\tau}$ is proposed here in context of algebraic analysis \cite{DPR}. The linear operator $D_{\sigma}^{\tau}$, specified with the help of two fixed maps $\sigma\;, \tau\colon M\rightarrow M$, generalizes the quantum derivative operator used in $h$- or $q$-calculus \cite{kac}. In the domain of $D_{\sigma}^{\tau}$ there are special elements defined as $D_{\sigma}^{\tau}$-polynomials and the corresponding Taylor formula is proved.