Abstract
The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.
Highlights
This triangle has some simple yet interesting properties that are familiar to most introductory algebra students: i) Horizontal rows add to powers of 2, which can, be shown by putting a= b= 1 in the binomial formula
There are three convenient ways of doing this: a) As a lower triangular matrix L where the binomial coefficients are placed in rows
B) As an upper triangular matrix U where the binomial coefficients are placed in columns
Summary
How to cite this paper: Izmirli, I.M. (2015) An Algorithm to Generalize the Pascal and Fibonacci Matrices. This triangle has some simple yet interesting properties that are familiar to most introductory algebra students: i) Horizontal rows add to powers of 2, which can, be shown by putting a= b= 1 in the binomial formula. The binomial coefficients, but the addition rule, which, is needed to generate the coefficients, were known to Indian mathematicians. The triangle was known in China in the early 11th century, a fact that is, according to [4], corroborated by the works of the Jia Xian (1010-1070) and Yang Hui (1238-1298). There are three convenient ways of doing this: a) As a lower triangular matrix L where the binomial coefficients are placed in rows. B) As an upper triangular matrix U where the binomial coefficients are placed in columns. See [6] for a proof
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