On $k$-Mersenne and k-Mersenne-Lucas Octonions

TL;DR

This paper introduces $k$-Mersenne and $k$-Mersenne-Lucas octonions, providing their closed-form formulas, identities, and generating functions, extending known properties of Mersenne octonions.

Contribution

It presents the first comprehensive study of $k$-Mersenne and $k$-Mersenne-Lucas octonions, including explicit formulas and identities, generalizing previous work on Mersenne octonions.

Findings

Closed-form formulas for $k$-Mersenne and $k$-Mersenne-Lucas octonions

Derivation of identities like Cassini, Catalan, Vajda

Generating functions for these octonions

Abstract

This paper aims to introduce the $k$-Mersenne and $k$-Mersenne-Lucas octonions. We give the closed form formulae for these octonions and obtain some well-known identities like Cassini's identity, d'Ocagne's identity, Catalan identity, Vajda's identity and generating functions of them. As a consequence k=1 yields all the above properties for Mersenne and Mersenne-Lucas octonions.