Hessenberg matrices and the generalized Fibonacci-Narayana sequence

Abstract

In this note, we define the generalized Fibonacci-Narayana sequence {G n (a, b, c)} n∈N. After that, we derive some relations between these sequences, and permanents and determinants of one type of upper Hessenberg matrix. with initial conditions G 0 (a, b, c) = 0, G i (a, b, c) = 1, for i = 1, 2,. .. , c − 1. The constants a and b are nonzero real numbers. We call this sequence generalized Fibonacci-Narayana sequence. Note that, if a = b = 1 and c = 2, the Fibonacci sequence is obtained, and if a = 1 = b and c = 3, the Narayana sequence is obtained [1, 12]. Other particular cases of the sequence {G n (a, b, c)} n∈N are • If a = b = 1, the generalized Fibonacci sequence is obtained [2]. G n = G n−1 + G n−c. • If c = 2, the generalized Fibonacci sequence is obtained. G n = aG n−1 + bG n−2. • If a = k, b = 1 and c = 2, the k-Fibonacci sequence is obtained [10]. F k,n = kF k,n−1 + F k,n−2 .

Some permanents of Hessenberg matrices

“…Matrix methods are useful tools for derivation some properties of linear recurrences. Some authors obtained many connections between certain sequences and permanents of Hessenberg matrices in the literature [1,[3][4][5][8][9][10]. The permanent of an n × n matrix A n = a i j is defined by…”

“…In [10], Ramírez showed some relations between the generalized Fibonacci-Narayana sequence and permanent of one type of upper Hessenberg matrix. For example,…”