On the fundamental theorem of $(p,q)$-calculus and some $(p,q)$-Taylor formulas

TL;DR

This paper explores the properties of the $(p,q)$-calculus, introduces polynomial bases, and presents new $(p,q)$-Taylor formulas along with the fundamental theorem and integration by parts within this calculus framework.

Contribution

It introduces new polynomial bases for $(p,q)$-derivatives and derives novel $(p,q)$-Taylor formulas and fundamental theorems, expanding the theoretical foundation of $(p,q)$-calculus.

Findings

Two polynomial bases for $(p,q)$-derivative are established.

New $(p,q)$-Taylor formulas for polynomials are derived.

Fundamental theorem of $(p,q)$-calculus and integration by parts formula are proved.

Abstract

In this paper, the $(p,q)$-derivative and the $(p,q)$-integration are investigated. Two suitable polynomials bases for the $(p,q)$-derivative are provided and various properties of these bases are given. As application, two $(p,q)$-Taylor formulas for polynomials are given, the fundamental theorem of $(p,q)$-calculus is included and the formula of $(p,q)$-integration by part is proved.