Cassini, d'Ocagne, and Vajda identities for n-step Fibonacci numbers

TL;DR

This paper extends classical Fibonacci identities such as Cassini, d'Ocagne, Catalan, and Vajda to the broader context of n-step Fibonacci numbers using the concept of n-determinants.

Contribution

It introduces the extension of well-known Fibonacci identities to n-step Fibonacci numbers through the framework of n-determinants.

Findings

Cassini identity extended to n-step Fibonacci numbers

d'Ocagne identity generalized for n-step Fibonacci sequences

Vajda identity adapted for n-determinants in n-step Fibonacci context

Abstract

Results of this paper concern $n$-determinants which we defined in the paper \cite{jan}. In the paper \cite{jabo}, $2$-determinants are considered. In this paper, we extend results from \cite{jabo} on $n$-determinants by proving that Cassini, d'Ocagne, Catalan and Vajda identities may be extended to hold for $n$-step Fibonacci numbers.