Abstract
In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is finite sums of reciprocals of Fibonacci numbers. We obtain spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers.
Highlights
The harmonic numbers have many applications in combinatorics and other areas
Where n k denotes the Stirling number of the first kind, counting the permutations of n elements that are the product of k disjoint cycles
In this paper, inspired by the definition of a harmonic number, we introduce harmonic Fibonacci numbers and give various identities for these numbers by using the difference operator
Summary
The harmonic numbers have many applications in combinatorics and other areas. Many authors have studied these numbers. The nth harmonic number, Hn, is defined by n. In [ ], Conway and Guy defined the nth hyperharmonic number of order r, Hn(r) for n, r ≥ by the following recurrence relations: Hn(r) =. The Fibonacci sequence is defined by the following recurrence relation, for n ≥ : Fn+ = Fn+ + Fn with F = , F =. In [ ] the authors use a property of the finite difference operator to show the validity of the identity ( ) as follows. We obtain the spectral and the Euclidean norms of circulant matrices with harmonic and hyperharmonic Fibonacci numbers. We state some theorems related to harmonic Fibonacci numbers. Theorem Let Fn be the nth harmonic Fibonacci number and m be a nonnegative integer.
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