On some new families of k-Mersenne and generalized k-Gaussian Mersenne numbers and their polynomials

TL;DR

This paper introduces new families of generalized k-Mersenne and Gaussian Mersenne numbers and polynomials, exploring their properties, identities, and relations to classical Mersenne numbers.

Contribution

It defines novel generalized k-Mersenne and Gaussian Mersenne numbers and polynomials, deriving formulas, identities, and exploring their mathematical properties.

Findings

Derived Binet, Cassini, and D'Ocagne identities for Gaussian Mersenne numbers

Established relations between generalized and classical Mersenne numbers

Introduced generalized Mersenne polynomials and studied their properties

Abstract

In this paper, we define new generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and the relation with classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials.