On Dual-Complex Numbers with Generalized Fibonacci and Lucas Numbers Coefficients

Abstract

In this paper, dual-complex Fibonacci numbers with generalized Fibonacci and Lucas coefficients are defined. Generating function is given for this number system. Binet's formula is obtained by the help of this generating function. Then, well-known Cassini, Catalan, d'Ocagne's, Honsberger, Tagiuri and other identities are given for this number system. Finally, it is seen that the theorems and the equations which are obtained for the special values p = 1 and q = 0 correspond to the theorems and identities in [2].

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

“…Hence, Section 3 is directly linked to the paper [36] for p = −1 (regarding dual-complex case) and the paper [38] for p = 1 (regarding dual-hyperbolic case). Additionally, Section 3 is closely associated with the papers [37], [41] and [39,40] regarding dual-complex, hyper-dual and dual-hyperbolic situations as a special case. This classification can be seen in Table 4.…”

“…By using dual-complex numbers in [14], dual-complex Fibonacci and Lucas numbers are defined in [36] and Binet's formulas, and D'Ocagne, Catalan's and Cassini's identities are obtained. 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].…”