h-analogue of Fibonacci Numbers

TL;DR

This paper introduces the h-analogue of Fibonacci numbers in a non-commutative h-plane, deriving identities, a Binet's formula, and a generating function, extending classical Fibonacci concepts to a new algebraic setting.

Contribution

The paper presents the first formulation of h-Fibonacci numbers in a non-commutative setting, including identities, Binet's formula, and generating functions.

Findings

h-Fibonacci numbers reduce to classical Fibonacci numbers when h=0 or h'=1.

Derived identities for h-Fibonacci numbers.

Established Binet's formula and generating function for h-Fibonacci numbers.

Abstract

In this paper, we introduce the h-analogue of Fibonacci numbers for non-commutative h-plane. For h h'= 1 and h = 0, these are just the usual Fibonacci numbers as it should be. We also derive a collection of identities for these numbers. Furthermore, h-Binet's formula for the h-Fibonacci numbers is found and the generating function that generates these numbers is obtained.