(q,h)-analogue of Newton's binomial formula

TL;DR

This paper introduces a new (q,h)-analogue of Newton's binomial formula within a deformed quantum plane, unifying and extending existing q- and h-analogues, and explores its properties for potential applications in (q,h)-deformed analysis.

Contribution

It presents the first derivation of the (q,h)-binomial formula, generalizing previous q- and h-analogues and laying groundwork for (q,h)-special functions and analysis.

Findings

Derived the (q,h)-binomial formula in the deformed quantum plane.

Showed reduction to classical, q-, and h-analogues under specific limits.

Explored properties of (q,h)-binomial coefficients.

Abstract

In this letter, the (q,h)-analogue of Newton's binomial formula is obtained in the (q,h)-deformed quantum plane which reduces for h=0 to the q-analogue. For (q=1,h=0), this is just the usual one as it should be. Moreover, the h-analogue is recovered for q=1. Some properties of the (q,h)-binomial coefficients are also given. This result will contribute to an introduction of the (q,h)-analogue of the well-known functions, (q,h)-special functions and (q,h)-deformed analysis.