Abstract

Binomial coefficients have long been studied by several mathematicians for several centuries [1] and they are currently grouped in what is called Pascal's triangle [2]. These coefficients are very useful not only in combinatorics, but also, they intervene in many fields such as enumeration, development of the binomial in algebra, development in series, and in probability distributions and statistics [3]. In addition, Pascal's formula, generative of these coefficients, leads us to think of a generalization of these coefficients and of the other strongly related mathematical tools. In this article, we will try to generalize the binomial coefficients, the Newton’s binomial as well as the Fibonacci sequence and establish their expressions. We will call these generalizations p-nomial coefficients, Atta’s p-nomial and p-bonacci sequence.

1. Introduction

Binomial coefficients Newton's binomial and the Fibonacci sequence have always fascinated not only mathematicians, but also lovers of nature and the fine arts. Countless works and research have been devoted to these themes, which have contributed to the study of other related subjects not only in mathematics but also in other fields such as physics, chemistry, computer science, drawing and painting as well as the construction fields ⁴.

The history of these three notions of mathematics is very long and dates back several centuries. Indeed, these tools have been studied by several mathematicians in India, Persia, China (Zhu Shijie and Yang Hui), Maghreb, Germany and Italy ¹.

We also know that there are strong links between the Fibonacci sequence, the binomial coefficients and the Newton’s binomial that can be established using analysis techniques such as recurrent sequences, polynomials and descending and ascending diagonals from Pascal's triangle. But we can also use the techniques of matrix calculation ⁴. Indeed, the binomial coefficients can be obtained by calculating the powers of a strictly triangular matrix having just the non-zero diagonal and sub-diagonal whose coefficients are all equal to 1.

In the same way, we will try to generalize the binomial coefficients under the name of p-nominal coefficients using the technique of matrix calculation; which will allow us to introduce a generalization of Pascal's triangle ³ called staircase of p-nominal coefficients. Note that these coefficients and multinomial coefficients should not be mixed. However, there is an obvious relation between these two notions.

At this stage, we will hope to introduce a concept similar to Newton's binomial called Atta’s p-nomial or p-nomial identity which should not be mixed with the multinomial (linked to the multinomial coefficients). Thereafter, the generalization of the Fibonacci sequence known as the p-bonacci sequence will be accessible and the establishment of its expression will be done using the staircase of the p-nomial coefficients by the same method followed to establish the expression of the Fibonacci sequence from Pascal's Triangle.

We will finally present some applications and establish some useful formulas using these tools such as the introduction of a new probability distribution (generalization of Binomial Distribution) call p-nomial distribution and new polynomial similar to Bernstein's polynomial ⁵ called p-Bernstein polynomial (Writing in italic) without forgetting that they will have other applications in other fields.

2. Definition of p-nomial Coefficients

Let be a non-zero natural integer. We define p-nomial coefficient and we note the k^th p-nomial coefficient among by the following recurrent relation:

Such that ⁶.

Example. For and

3. Properties of p-nomial Coefficients

Property 1

The -nomial coefficients verify the following property:

Proof. Using recurrence on n. For , it’s trivial. For , for all (by definition).

We Suppose by recurrence that:

For , we have:

This completes demonstration by recurrence.

This property allow us to build a staircase each line of which consists of elements numbered from to that we call the staircase of p-nomials. For , it’s exactly Pascal’s triangle.

The number is called staircase step.

Property 2

The -nomial coefficients verify the following property:

Proof. Using recurrence on n.

For , it’s trivial. For :

(by definition).

Suppose by recurrence that, for

For , we have:

This completes demonstration by recurrence.

Property 3

we have:

Proof. Using recurrence:

For , and (by definition because ). Now, we suppose by recurrence that, for

For , we have:

This completes demonstration by recurrence.

1-nomial coefficients (uninomial)

For the case , the 1-nomial coefficients are given by:

2-nomial coefficients (binomial)

For the case , the 2-nomial coefficients are given by:

We easily notice that the sequence verify relation of the binomial coefficients ²:

We thus deduce that:

Therefore, 2-nomial coefficients are exactly the binomial coefficients.

4. Fundamental Theorem

Introduction

denote the integer part of the real number

The p-nomial coefficients are obtained by calculation of the matrix where:

This matrix is called -unit matrix of size .

Indeed, the non-zero coefficients of the columns of this matrix are identical and therefore we will be interested in the first column of this matrix. The q^th nonzero coefficient in this column represents the . To find a relation between and , proceed as follows. Let's write the -unit matrix of size n as:

Where:

So, we will have:

Well, the first column of the strictly triangular matrix contains non-zero elements, which means that contains the p-nomial coefficients . So, we can conclude that the are combinations of with as coefficients of sum. The line of matrix contains the p-nomial coefficients:

Identity of complements

We have the following identity:

which means that the sum is independent of p.

Proof. If then:

If then and we have:

Hence the validity of the identity. This will help us to write the expression of the recurrent relation linking -nomial and p-nomial coefficients.

Lemma 1

Let a natural number such as its Euclidean division by is written . We have:

Proof. Let a non-zero natural integer such that . Because of , we get:

So, we can write:

Lemma 2

Let and four natural numbers such as and . We have the following equality:

Proof. Indeed, according to the definition formula of p-nomials coefficients:

But, , so:

Fundamental theorem

Let

The -nomial coefficients and the p-nomial coefficients are linked by the following recurrent relation called p-nominal formula:

Proof. The proof of the fundamental theorem of p-nomials is based on a proof by recurrence. We will thus use recurrence many times. To initiate the recurrence, let us demonstrate this formula for the first value of p. For :

So, it’s true for . Now suppose that for , we have:

Now, let's prove the recurrence for . Using definition formula and recurrence hypothesis, we can write that:

(1)

The first proposition of demonstration consists in treating the possible cases according to the Euclidean division of by and for each case, treat different possible cases by comparing the numbers and : , or .

We need also to use a lot lemmas 1 and 2. We note the Euclidean division of by as follows:

but this demonstration is very long since it discusses several cases. We can use a more concise demonstration which consists in noticing that in formula , we can sum from the minimum of the index which is to the maximum which is since the terms which will be added are null. We should also notice that:

For that, we’ll discuss just two cases:

First case: If and:

Then, we get:

Using and , we conclude that:

Second case: If :

and:

We obtain thus:

Using , we conclude that:

We thus treated all cases; which completes the proof of the fundamental theorem of p-nomials by recurrence.

Expressions of p-nomial coefficients

Using the fundamental theorem, we can establish an expression for the p-nomial coefficients. We give below the expression of -nomials:

To simplify the writing, we introduce the symbol to group the summations and order them by an index that varies in a given interval. For :

We can also notice that:

It’s a multinomial coefficient ⁷.

5. Atta’s p-nomial

Definition

Let and two complex numbers and a non-zero natural integer. We define Atta’s -nomial by:

p-nomial theorem

We have following equality:

Proof. Using recurrence. For , we get: .

For , we get:

Let suppose that for

For , we have:

If we consider , then . For a fixed , varies from to and varies from to . So, by changing the form of writing, we can write:

which completes the proof of p-nomial theorem by recurrence.

6. p-bonacci Sequence

Definition

We define a generalization of the Fibonacci sequence (which is the 2-bonacci) ^{1, 6, 8}. p-bonnaci is defined by:

Expression

Let be a natural integer. The n^th number in the sequence of p-bonacci is given by the formula:

Proof. the demonstration is done by recurrence on n using the Euclidean division of n by p and by discussing according to the rest of the division r, the three cases , et . Furthermore, it is assumed by recurrence that:

For example, Tetra-Bonacci sequence is given by using the staircase of tetra-nomials as follow :

We will give a more detailed example at the end of this article.

Golden numbers sequence

We define the golden ratio of order p as the limit of:

The sequence is called golden numbers sequence.

Indeed, we can prove that the golden ratio exist using recurrence relation . In addition, we have ⁹:

And for the golden number is the unique positive solution different from 1 of the equation:

We can easily demonstrate using the intermediate value theorem that:

So, we can deduce that:

In addition, one can establish an asymptotic expression with the golden ratio as follows. We note this expression:

Since we have

By replacing by the expression of

By proceeding with a limited development:

We thus obtain and:

7. Trinomial Coefficients

Definition

The k^th trinomial coefficient among noted verify following recurrence relation:

Such that .

Trinomial staircase (triangle)

It is easy to construct triangle of trinomials. Indeed, we follow the same approach of construction of the Pascal’s triangle:

"We sum the three successive elements of line n and put it below the central element of the three elements summed for line number n +1":

The triangle above is the triangle of trinomials up to .

Atta’s trinomial

For all complex numbers and :

Expression of trinomial coefficients.

Let

The trinomial coefficients and the binomial coefficients are linked by the following recurrent relation:

Properties 1

Proof. By using Atta’s trinomial for .

Property 2

Let p a prime number. We have following properties:

Proof. By using the expression of trinomial coefficients:

Since is prime, , then doesn’t divide . Also, we have: , which give:.

If then . Because of we deduce that doesn’t divide . If then and this is not possible because we deduce that doesn’t divide . Thus, doesn’t divide and we deduce that:

Applications

Formulas

Using the p-nomial, we can have several formulas among which we cite:

p-Bernstein polynomial

For a degree , there are p-Bernstein polynomials, defined over the interval , by:

These polynomials form the basis of the vector space of polynomials of degrees less than or equal to mp.

These polynomials verify the following properties:

where Bernstein polynomial ⁵.

p-nomial distribution

Definition

Let and two non-zero natural integers. A random variable follow a nomial distibution (generalization of binomial distribution) ¹⁰, where if:

Indeed, this law is well defined:

Which shows that:

Moreover:

Esperance

Note

The Esperance of the p-nominal law is given by:

For , we have We find : this is Esperance of binomial law.

For , We find : this is Esperance of trinomial distribution.

Inverse of a triangular matrix

We consider the following triangular matrix:

The inverse of this matrix is given by:

Where:

References