Properties of hyperbolic third order Jacobsthal and Jacobsthal Lucas numbers

In this study, the general properties of Jacobsthal and Jacobsthal-Lucas number sequences, which were defined by Horadam in 1996, were examined, and the definitions and some important properties of second order hyperbolic Jacobsthal and Jacobsthal-Lucas and third order hyperbolic Jacobsthal and Jacobsthal-Lucas sequences were given. In the first chapter, the general history of number sequences and definitions of some important number sequences are given. In the second part, definitions of second order Jacobsthal and Jacobsthal-Lucas number sequences are given and some properties of these sequences are given. Then, second order hyperbolic Jacobsthal and hyperbolic Jacobsthal-Lucas number sequences are defined, Binet formula, generating function, Simson formula and linear sum formulas are given. In the third chapter, first the definitions and basic properties of the third order Jacobsthal and Jacobsthal-Lucas number sequences are given, then the Binet formulas, generating functions and some identities of the third order hyperbolic Jacobsthal and hyperbolic Jacobsthal-Lucas number sequences are obtained. Finally, at the end of this chapter, hyperbolic Jacobsthal vectors are introduced and the Lorentzian inner product of these vectors is studied.