On the generalized fibonacci and Lucas 2^k−ions

Abstract

This study introduces the modified generalized Fibonacci and Lucas 2^k−ions whichare the generalizations of several quaternions, octonions and higher order dimensional algebras.We give the generating functions, the Binet formulas and well-known identities such as Catalan’sidentity and Cassini’s identity for the modified generalized Fibonacci and Lucas 2^k−ions.