Generalized Fibonacci Sequences

The Fibonacci sequence occurs everywhere in nature. This sequence has applications to distinct branches of mathematics, as well as varied situations of the world. The Fibonacci sequence has applied to define different algebraic structures such as Fibonacci quaternion, Fibonacci octonion, Fibonacci sedenion, Fibonacci group, Fibonacci graph, Fibonacci lattice etc. Such a theory recently received much attention and now constitutes a great branch in number theory. This book aims to extend the idea of coupled Fibonacci sequences of lower order to the multiplicative coupled Fibonacci sequence of rth order. The various properties of these sequences have been proved under two particular schemes. The binomial sums and congruence properties have been investigated for k-Fibonacci and k-Lucas sequences. The matrices E and M are defined for k-Fibonacci and k-Lucas sequences. These matrices have been used to establish different properties of k-Fibonacci and k-Lucas sequences using matrix methods. Furthermore, the hyperbolic k-Fibonacci quaternions and hyperbolic k-Fibonacci octonions have been defined and different properties of quaternions and octonions have been studied by various techniques.