h analogue of Newton's binomial formula

TL;DR

This paper introduces the $h$-analogue of Newton's binomial formula within the $h$-deformed quantum plane, generalizing classical binomial coefficients and exploring their properties for potential applications in $h$-deformed analysis.

Contribution

The paper derives the $h$-analogue of Newton's binomial formula in the $h$-deformed quantum plane and examines its properties, extending classical binomial concepts.

Findings

$h$-binomial coefficients reduce to classical form when $h=0$.

For $h=1$, $h$-binomial coefficients become factorial ratios.

Properties of $h$-binomial coefficients are established.

Abstract

In this letter, the $h$--analogue of Newton's binomial formula is obtained in the $h$--deformed quantum plane which does not have any $q$--analogue. For $h=0$, this is just the usual one as it should be. Furthermore, the binomial coefficients reduce to $\frac{n!}{(n-k)!}$ for $h=1$. \\ Some properties of the $h$--binomial coefficients are also given. \\ Finally, I hope that such results will contribute to an introduction of the $h$--analogue of the well--known functions, $h$--special functions and $h$--deformed analysis.