Generalization of Gaussian third-order Jacobsthal numbers and their new families

TL;DR

This paper introduces a generalized family of Gaussian third-order Jacobsthal numbers with arbitrary initial values, explores their algebraic properties, and proposes a new k-generalized version with connections to existing sequences.

Contribution

It presents a novel generalization of Gaussian third-order Jacobsthal numbers and introduces the k-generalized version, expanding the understanding of these sequences and their properties.

Findings

Derived Binet's formula and generating functions.

Established identities like Cassini's and d'Ocagne's.

Connected the new sequences with existing Jacobsthal numbers.

Abstract

In this study, we introduce the generalized Gaussian third-order Jacobsthal numbers with arbitrary initial values and discuss two particular cases, namely, Gaussian third-order Jacobsthal and Gaussian modified third-order Jacobsthal numbers. In this paper we discuss several of its algebraic properties such as Binet's formula, partial sum, generating function, negative subscript elements, d'Ocagne's and Cassini's identities. Furthermore, we study and introduce a new generalization of this sequence called $k$-generalized Gaussian third-order Jacobsthal numbers. We present several of its properties and its connection with the generalized third-order Jacobsthal numbers.