Fundamental properties of extended Horadam numbers

Abstract

In this paper, the extended Horadam numbers are introduced by using dual-generalized complex, hyperbolic-generalized complex and complex-generalized complex numbers. Then, generating function, Binet's formula, D'Ocagne's, Catalan's and Cassini's identities are given. Moreover, specialy matrix representations of the extended Horadam numbers are investigated. In conclusion, the results and classification of the special cases are introduced.