On higher order Fibonacci hyper complex numbers. (English) Zbl 1485.11036
Summary: This paper deals with developing a new class of quaternions, octonions and sedenions called higher order Fibonacci \(2^m\)-ions (or-higher order Fibonacci hyper complex numbers) whose components are higher order Fibonacci numbers. We give recurrence relation, Binet formula, generating function and exponential generating function of higher order Fibonacci \(2^m\)-ions. We also derive some identities such as Vajda’s identity, Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity with the aid of the Binet formula. Finally, we develop some matrix identities involving higher order Fibonacci \(2^m\)-ions which allow us to obtain some properties of these higher order hyper complex numbers.
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
Keywords:
hypercomplex numbers; higher order Fibonacci \(2^m\)-ions; recurrence relations; generating functionsReferences:
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