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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.

The binomial formula

where the binomial coefficients

for any two non-negative integers

It is customary to call the triangular array made up of the binomial coefficients

the Pascal’s triangle. 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, of course, easily be shown by putting

ii) The horizontal rows represent powers of 11, which can, of course, easily be shown by putting a = 10 and b = 1 in the binomial formula.

iii) Adding any two successive numbers in the diagonal containing the triangular numbers

iv) In the expansion of

Any coefficient is of the form

It is now acknowledged that this triangle was known well before Blaise Pascal (1623-1662) who “introduced” it in his famous 1653 treatise, Traité du triangle arithmétique. Indeed, not only the binomial coefficients, but in fact, the addition rule, which, of course, is needed to generate the coefficients, were known to Indian mathematicians^{1}. For instance, according to Edwards [^{th} century CE, in the book Meru-prastaara^{2} by Halayudha (?-?). See [

Persian mathematicians were also well acquainted with the binomial coefficients―this can be seen, for example, in the writings of Al-Karaji (953-1029) and later in those of Omar Hayyam (1048-1131), who indeed set up the entire triangle. Thus, some scholars and historians refer to the triangle as the Khayyam-Pascal triangle (see [

Many other cultures were familiar with the triangle and its properties as well. For example, the triangle was known in China in the early 11^{th} century, a fact that is, according to [

There were also precedents in the west. The German humanist, Petrus Apianus (1495-1552), known for his works in mathematics, astronomy, and cartography, published the full triangle in 1527. In the second half of 16^{th} century, parts of the triangle were published by the German monk and mathematician Michael Stifel (1487- 1567), and the Italian mathematicians Niccolo Fontana Tartaglia (1499-1557) and Gerolamo Cardano (1501- 1576) See [

A closely related idea is that of the Pascal matrix. The Pascal matrix is an infinite matrix containing the Pascal triangle as a submatrix. There are three convenient ways of doing this:

a) As a lower triangular matrix

b) As an upper triangular matrix

c) As a symmetric matrix

See [

Clearly,

See [

As is well-known, the Fibonacci sequence

and for

The sequence is named after Leonardo of Pisa (Fibonacci) (c.1170-c. 1250), who in his 1202 book Liber Abaci introduced it to the European readers. However, as was the case with Pascal’s triangle, this sequence had been described earlier by Indian mathematicians as well. See [

The Fibonacci triangle is a two-dimensional version of the Fibonacci sequence. It is defined as follows:

For

and for

So this is a triangle with Fibonacci sequences on the sides. Note that the subdiagonals are Fibonacci sequences as well, except that the starting value is no longer . So, the left edge of the triangle (as well as the right edge) is the Fibonacci sequence, the diagonal parallel to it is the Fibonacci sequence, the next diagonal is the Fibonacci sequence starting with

See [

It is easy to see that if we let

for

arranged in a matrix gives us the Pascal matrix. For,

If we now form a matrix where the

It is now natural to ask the following question: If

Definition 1. Let

whose

Example 1. The simplest example is

So, the power series matrix generated by

Example 2. As another example let us consider the function

by

implying

and

for

To find power series expansions of

Hence, differentiating both sides of the identity for

Similarly,

and so on. Consequently,

Hence, the power series matrix generated by

Now we want to give an algorithm that will give us the entries of

series expansion of

Set

and if

Now for

and

For

and

To see why this algorithm works, for

Note that the coefficient of

is

Consequently, for

Since

we have for

This algorithm is, of course, a natural generalization of the addition process we apply to calculate various coefficients in Pascal’s triangle. In fact, in case

Examples:

1) The power series matrix of

2) The power series matrix of