Институт Золотого Сечения — Математика Гармонии

Аlexey P. Stakhov
Fibonacci matrices, a generalization of the «Cassini formula», and a new coding theory

International Club of the Golden Section,
6 McCreary Trail, Bolton, ON, Canada L7E2C8

Abstract

We consider a new class of square Fibonacci (p + 1) × (p + 1)-matrices, which are based on the Fibonacci p-numbers (p = 0,1,2,3,...), with a determinant equal to +1 or — 1. This unique property leads to a generalization of the «Cassini formula» for Fibonacci numbers. An original Fibonacci coding/decoding method follows from the Fibonacci matrices. In contrast to classical redundant codes a basic peculiarity of the method is that it allows to correct matrix elements that can be theoretically unlimited integers. For the simplest case the correct ability of the method is equal 93.33% which exceeds essentially all well-known correcting codes.

1. Introduction

In the last decades the theory of Fibonacci numbers [1,8] was complemented by the theory of the so-called Fibonacci Q-matrix [1,2]. The latter is a square 2×2 matrix of the following form:
 (1)

In [1] the following property of the nth power of the Q-matrix was proved
 (2)
 (3)

where n = 0,±1,±2,±3,..,Fn-1,Fn,Fn+1 are Fibonacci numbers given with the following recurrence relation:
 (4)

with the initial terms
 (5)

Note that identity (4) is called Cassini formula in honor of the well-known 17th century astronomer Giovanni Cas-sini (1625-1712) who derived this formula.

In 1977 the author introduced so-called Fibonacci p-numbers [3]. For a given integer p = 0,1,2,3,... the Fibonacci p-numbers are given with the following recurrence relation:
 (6)

with the initial terms
 (7)

In [4] the notion of the Qp-matrices (p = 0,1,2,3,...) was introduced. This notion is a generalization of the Q-matrix (1) and is connected to the Fibonacci p-numbers (6) and (7).

The main purpose of the present article is to develop a theory of the Qp-matrices. The next purpose is to give a generalization of the «Cassini formul»a (4) that follows from the theory of the Qp-matrices. Also a new approach to a coding theory, which is based on the Qp-matrices, is considered.