Fibonacci Fervour in Linear Algebra and Quantum Information Theory

Chatterjee, Manami; Singh, Ajit; Sivakumar, K. C.

doi:10.48550/arxiv.1804.08100

Cited by 1 publication

(1 citation statement)

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“…We have seen that the Fibonacci sequence can be generalized in many ways such as generalizations of the Euclid's theorem, the recurrence relations, and the characteristic equations. There is yet another way to study and generalize the Fibonacci sequence and derive many interesting properties of these numbers using a matrix representation [84][85][86][87][88][89][90][91][92][93][94][95]. By matrix methods, while Silvester [86] derived many interesting properties of the Fibonacci numbers, Kalman [87] generalized Fibonacci numbers.…”

Section: Q-matrix and Determinantal Identitymentioning

confidence: 99%

Additive Sequences, Sums, Golden Ratios and Determinantal Identities

Kumar¹

2021

Preprint

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The Fibonacci sequence is a series of positive integers in which, starting from 0 and 1, every number is the sum of two previous numbers, and the limiting ratio of any two consecutive numbers of this sequence is called the golden ratio. The Fibonacci numbers and the golden ratio are two significant concepts that keep appearing everywhere. In this article, we investigate the following issues:• We recall the Fibonacci sequence, the golden ratio, their properties and applications, and some early generalizations of the golden ratio. The Fibonacci sequence is a 2-sequence because it is generated by the sum of two previous terms,As a natural extension of this, we introduce several typical p-sequences where every term is the sum of p previous terms given p initial values called seeds. In particular, we introduce the notion of 1-sequence. We then discuss generating functions and limiting ratio values of p-sequences. Furthermore, inspired by Fibonacci's rabbit pair problem, we consider a general problem whose particular cases lead to nontrivial additive sequences.• We establish a family of determinantal identities of which the Cassini's identity is a particular case.

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