Some properties of deformed $q$-numbers

TL;DR

This paper explores properties of deformed q-numbers within nonextensive statistical mechanics, introducing new algebraic operations, a novel q-product, and various q-Pascal's triangle patterns to deepen understanding of q-deformed structures.

Contribution

It introduces a new q-product that distributes over the q-sum and presents different patterns of q-Pascal's triangles based on q-numbers.

Findings

A new q-product that distributes over q-sum.

Patterns of q-Pascal's triangles based on q-numbers.

Implications for deformed algebraic structures in nonextensive mechanics.

Abstract

Nonextensive statistical mechanics has been a source of investigation in mathematical structures such as deformed algebraic structures. In this work, we present some consequences of $q$-operations on the construction of $q$-numbers for all numerical sets. Based on such a construction, we present a new product that distributes over the $q$-sum. Finally, we present different patterns of $q$-Pascal's triangles, based on $q$-sum, whose elements are $q$-numbers.