On Dual-Hyperbolic Numbers with Generalized Fibonacci and Lucas Numbers Components

Abstract

Dual-hyperbolic Fibonacci and Lucas numbers with Fibonacci and Lucas coefficients areintroduced by Cihan et al. and some identities and theorems are given regarding modulesand conjugates of these numbers. Later, generating function and Binet’s formula with thehelp of this generating function have been derived. Also, Binet formula, Cassini’s, Catalan’s,d’Ocagne’s, Honsberger and Tagiuri identities are found for dual-hyperbolic numbers withgeneralized Fibonacci and Lucas coefficients. While these operations are being done, wewill benefit from the well-known Fibonacci and Lucas identitites. Moreover, it is seen thatthe results which are obtained for the values p = 1 and q = 0 corresponds to the theoremsin the article by Cihan et al. [1].