
Abstract
We consider a new class of square Fibonacci (p + 1) × (p + 1)matrices, which are based on the Fibonacci pnumbers (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 wellknown correcting codes.
1. Introduction
In the last decades the theory of Fibonacci numbers [1,8] was complemented by the theory of the socalled Fibonacci Qmatrix [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 Qmatrix was proved
(2) 
(3) 
where n = 0,±1,±2,±3,..,F_{n1},F_{n},F_{n+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 wellknown 17th century astronomer Giovanni Cassini (16251712) who derived this formula.
In 1977 the author introduced socalled Fibonacci pnumbers [3]. For a given integer p = 0,1,2,3,... the Fibonacci pnumbers are given with the following recurrence relation:
(6) 
with the initial terms
(7) 
In [4] the notion of the Q_{p}matrices (p = 0,1,2,3,...) was introduced. This notion is a generalization of the Qmatrix (1) and is connected to the Fibonacci pnumbers (6) and (7).
The main purpose of the present article is to develop a theory of the Q_{p}matrices. The next purpose is to give a generalization of the «Cassini formul»a (4) that follows from the theory of the Q_{p}matrices. Also a new approach to a coding theory, which is based on the Q_{p}matrices, is considered.
See full text of this article at PDF format (202Kb).