NON-NEWTONIAN JACOBSTHAL AND JACOBSTHAL-LUCAS NUMBERS: A NEW LOOK

Abstract

In this study, we introduce a novel version of Jacobsthal and Jacobsthal-Lucas numbers, termed as non-Newtonian Jacobsthal and non-Newtonian Jacobsthal-Lucas numbers. We investigate various characteristics of these newly defined sequences. Additionally, we explore several formulas and identities such as Cassini’s identity, d’Ocagne’s identity, Binet’s formula, Gelin-Ces`aro’s identity, Honsberger’s identity, and Melham’s identity associated with these new types. Furthermore, we find the generating functions for such sequences. The novel feature of this study is to generalize the notions of Jacobsthal numbers by using non-Newtonian calculus. If we take the identity function I instead of the generator α in the construction of non-Newtonian Jacobsthal numbers, then non-Newtonian Jacobsthal numbers turn into the classical Jacobsthal numbers, so our results in this paper improve and generalize the known corresponding results in the literature.