On the (r, p, k)-Generalized Jacobsthal Numbers and the Jacobsthal Fundamental Fibonacci System

The main goal of this paper is to study the model of the sequences of (r, p, k)-generalized Jacobsthal numbers, by the approach based on the properties of its associated Jacobsthal fundamental Fibonacci system and its fundamental sequence. Some linear and combinatorial properties are established. Moreover, the related matrix formulation allows us to provide new identities, which are declined in combinatorial form. Especially, the linear Jacobsthal Cassini identity and its combinatorial formulation are furnished. Finally, illustrative special cases and examples are given.