Simson Identity of Generalized m-step Fibonacci Numbers

TL;DR

This paper extends the classical Simson identity to generalized m-step Fibonacci sequences, providing new formulas and identities that broaden understanding of Fibonacci generalizations.

Contribution

It introduces a generalized Simson identity for m-step Fibonacci numbers and presents specific identities for particular cases, advancing Fibonacci sequence theory.

Findings

Derived a generalized Simson identity for m-step Fibonacci numbers

Presented specific identities for particular generalized sequences

Enhanced understanding of Fibonacci sequence generalizations

Abstract

One of the best known and oldest identities for the Fibonacci sequence $F_n$ is $F_{n+1}F_{n-1}-F_{n}^2=(-1)^n$ which was derived first by R. Simson in 1753 and it is now called as Simson or Cassini Identity. In this paper, we generalize this result to generalized m-step Fibonacci numbers and give an attractive formula. Furthermore, we present some Simson's identities of particular generalized m-step Fibonacci sequences.