Generalization of Gaussian Mersenne numbers and their new families

Abstract

In this article, we present the generalized Gaussian Mersenne numbers with arbitrary initial values and discuss two particular cases, namely, Gaussian Mersenne and Gaussian Mersenne-Lucas numbers. We present their various algebraic properties such as Binet’s formula, negatively subscripted elements, Catalans’s, Cassini’s, and d’Ocagne’s identities, partial sum, binomial sum, generating and exponential generating functions, etc. In addition, we study a new generalized sequence arising from the explicit expression made with the characteristic roots and refer to them as the k-generalized Gaussian Mersenne numbers. We present various identities of them and show their connections with the generalized Gaussian Mersenne numbers.