A Study on Dual Hyperbolic Fibonacci and Lucas Numbers

Abstract

In this study, the dual-hyperbolic Fibonacci and dual-hyperbolic Lucas numbers are introduced. Then, the fundamental identities are proven for these numbers. Additionally, we give the identities regarding negadual-hyperbolic Fibonacci and negadual-hyperbolic Lucas numbers. Finally, Binet formulas, D’Ocagne, Catalan and Cassini identities are obtained for dual-hyperbolic Fibonacci and dual-hyperbolic Lucas numbers.

Two Generalizations of Dual-Hyperbolic Balancing Numbers

“…For special values of W n , we obtain the dual-hyperbolic Fibonacci numbers, the dual-hyperbolic Pell numbers, the dual-hyperbolic Jacobsthal numbers, the dual-hyperbolic balancing numbers, etc. Theorems 6-9 generalize the previously obtained properties of the dual-hyperbolic balancing numbers, some results of [17][18][19], and more specifically, the Binet-type formula, Catalan's identity, Cassini's identity, and d'Ocagne's identity for the dual-hyperbolic Fibonacci-type numbers.…”

“…It suffices to mention papers that have appeared recently. Cihan et al [17] introduced dual-hyperbolic Fibonacci and Lucas numbers. The dual-hyperbolic Pell numbers (quaternions) were introduced quite recently by Aydın in [18].…”

A comprehensive survey of dual-generalized complex Fibonacci and Lucas numbers

“…Likewise, dual-complex numbers with generalized Fibonacci and Lucas numbers coefficients are discussed in [37]. In analogy to dual-complex Fibonacci and Lucas numbers, dual-hyperbolic Fibonacci and Lucas numbers and their identities are introduced in [38]. Besides, dual-hyperbolic numbers with generalized Fibonacci and Lucas numbers coefficients are examined in [39,40].…”

Fundamental properties of extended Horadam numbers

“…For the special real values p = −1, p = 0 and p = 1, dual-complex, hyper-dual, dual-hyperbolic, hyperbolic-complex, bihyperbolic and bicomplex numbers are obtained from DGC, HGC and CGC numbers (see detailed classification in [12]). As we move from number systems to elements of Horadam sequences, some papers can be examined in literature, [2,4,7,11,13,14,24,28,33].…”