Abstract

Polynomials and matrices have played a very important role in the development of different branches of mathematics. Indeed, several mathematicians have introduced classical polynomials very useful for the scientific community such as the Lagrange’s interpolation polynomials, the Chebyshev’s polynomials and the Bernstein’s polynomials [1,2]. Also, there is a strong link between polynomials and matrices through the notions of the determinant, the characteristic polynomial and the minimal polynomial. Similarly, we will introduce in this article two polynomials which we will call twin polynomials as well as a matrix called kernels matrix. Finally, we will present some applications such as the resolution of recurrent sequences with second member and the establishment of several sum formulas.

1. Introduction

Polynomials and matrix calculus continue to attract the intension of researchers thanks to the simplifications they allow for the various scientific problems encountered. In addition, the strong links between these two branches make their use frequent. The twin polynomials introduced in this article show links with other classical mathematical tools such as binomial coefficients, trigonometric kernels, the Fibonacci sequence as well as recurrent sequences of order 3. The kernels matrix is a unitary matrix which will reverse several Hermitian matrices. We will then deduce several formulas and present some applications of these new tools.

2. Definitions

If is a complex number then the notation represents the absolute value of in the sense of complex numbers which is equal to:

We write thus:

To simplify, throughout the rest of this research, the following four complex-valued functions are defined while respecting the convention mentioned above:

we also note:

We define Factorial Mean by:

The factorial Mean verify the following relations:

Proof. For :

Let P a polynomial of two variables. We say that P is Even if:

We say that P is Odd if:

3. Twin Polynomials

For every recurrence relation, there is a unique polynomial of degree such that:

Proof. This is due to the setting of the initial conditions of the recurrence relations. Indeed, suppose there is another polynomial verify the same recurrence relation such as: and We consider the polynomial . We have . The polynomial verify the same recurrence relation. By recurrence, we show that: hence . We follow the same reasoning to demonstrate the unicity of .

The first expressions of and for :

Proof. By recurrence, for and :

For :

and have same parity of : and are even and and are odd.

Proof. It’s easy to prove it by recurrence.

All the monomials of and are of degree n.

Proof. It’s easy to prove it by recurrence.

The polynomials are given by:

Proof. We will present the demonstration for and then using the property 2, we can easily establish the demonstration for Using preceding lemmas, distinction is made according to the parity of . we note:

Now, using recurrence relation:

We get following system equations:

Using recurrence, we prove that: .

For , we have . We suppose that it’s true for . For :

Using variable change :

verify Factorial Mean Equation (FME). We conclude that: and we get:

Which completes the demonstration.

We can also check these formulas by using recurrence relation. For and : they are verified.

Twin polynomials and verify the following differential equations:

Proof. Let P polynomial solution of such that . We have:

Replacing in the equation , we find:

We get so the following system:

We conclude that:

Or even better:

For even numbers:

For odd numbers:

Now, to demonstrate the equation , we can easily use property 2 and equation :

We get:

The staircase of factorial means gathers in its steps the factorial means. The first column of the staircase represents the integer natural number n. Each step of the staircase contains two lines representing the factorial means for the even integers and for the odd integers in decreasing order: for the index step n (from the first column), the first line of the step represents the factorial means for 2n (k even varying from 2n to 0) and the second line represents the factor means for 2n+1 (k odd varying from 2n+1 to 1).

To build the staircase, we use the Factorial Mean Equation (MFE). To simplify the building procedure, we have put arrows on the stairs below. The oblique arrow means "+" (addition) and the downward vertical arrow means "=". The following staircase is by way of example constructed for n = 7:

Note. It should be noted that the factorial means staircase also contains the binomial coefficients of Newton's identity (the diagonals) ^{3, 4} as well as the elements of the Fibonacci sequence ⁵.

The Fibonacci’s sequence is defined by ⁵:

If we note and ⁶:

Then:

Proof. We have .

By proceeding by recurrence, we assume that: .

Which proves that: . For :

So . Using the characteristic equation , we obtain:

a) In the factorial means staircase, at step number n, the sum of the elements of the first line is equal to the number and the sum of the elements of the second line equal to the number .

b) In the factorial means staircase, the sum of all the elements of step number n is equal to .

c) We note the golden ratio. We have the following limits ⁵:

Let . The characteristic equations of sequences and are:

Ÿ Case 1: .

Ÿ Case 2: . We prove that:

Thus, we can write the expressions as follows:

Using the initial conditions, one can easily find the constants a, b, c and d. After solving the systems of equations, we obtain the following analytical expressions:

(1)

Generating function of polynomialis obtained as follow:

Generating function of polynomialis obtained as follow:

For , we can factorize twin polynomials:

In particular (2):

Proof. Using and by setting , we get:

Thus, the roots of the polynomial are given by:

The factorization of polynomial is:

Using property 2, we deduce easily:

Noticing that and , we deduce easily .

Using the different expressions of these two polynomials, we can derive several formulas and several sums. we cite as an example some:

Another expression of the Dirichlet kernel ⁷:

A formula verified by the Dirichlet kernels:

4. Kernels Matrix

We define trigonometric kernel by:

We define trigonometric kernel by:

Proving that by using formulas and , we deduce that:

We have the following properties:

We define the Kernels matrix of size n by:

Let . Kernel matrix is a orthogonal matrix ⁸.

Proof. We consider the matrix defined by:

Where . Let's calculate the determinant of this matrix noted for such that:

If we develop this determinant from the last column then the resulting determinant from the last row, we get:

we easily notice that:

In particular, if is a real number then:

By setting and , we obtain:

The matrix is invertible if and only if . This is a condition that is checked if:

Using the same previous approach, the characteristic polynomial of , after development, verify:

By setting and

we find:

Thus, the polynomial satisfies the same recurrence relation as the polynomial : ,

It is then easy to determine the spectrum of :

Now, we will look for the eigenvectors to diagonalize the matrix .

Let

the eigenvectors associated with the eigenvalues:

We proceed to determine the eigenvectors. For that, we consider the numerical sequence defined by:

We can easily prove that and therefore is well defined. We obtain the following system of equations:

The characteristic equation of this sequence is given by:

The reduced discriminant of this equation is:

And the two solutions are:

The sequence is therefore given by:

Using the initial conditions, after solving the system:

Then, we divide by the norm in order to normalize the eigenvector:

But for , we have and more:

It is the normalized kernel which is the component of index of the kernels matrix.

Now, let . Given that , will be symmetric real matrix ⁸, therefore its passage matrix will be orthogonal. Moreover, the eigenvectors are normalized; which allows us to conclude that the kernels matrix is orthonormal.

The determinant of the kernels matrix is called kernels determinant. It’s noted :

We can easily deduce that:

5. Applications

a) If we note the following determinant:

So:

b) For n sufficiently large, we have the following equivalence:

c) We have the following special case ⁹:

Where is the polynomial of Chebyshev of the 2^nd species. We can thus deduce several results on the factorial means using the polynomials of Chebyshev. We can also easily deduce a recurring relation verified by :

d) Example for kernels matrix:

e) Resolution of a recurrence sequence

Let and We consider the recurrent sequence defined by:

We complete this system by defining :

By writing this linear system in its matrix form, we get:

Where and are the column vectors of and (Last component of is ). We notice that is a Hermitian matrix ⁸. In this case, we have and . Also, the eigenvalues of the matrix are given by:

If we suppose that , then the matrix will be inversible. We note the following trigonometric kernels:

We use diagonalization formula to reverse :

Where is the diagonal matrix similar to . We obtain then:

The following sums are called arms:

We easily obtain the which are given by:

For , we deduce the expression of :

Then, we replace by its expression in order to obtain all the using the known data of the problem:

f) Particular case

If we suppose that :

Using the previous formula, we can write:

Now we use the classic method of the characteristic equation :

The reduced discriminant: The two solutions of this equation are:

The recurrent sequence is written in the form:

Where . To determine these constants, we use the initial conditions. We find:

Then, we obtain the expression of the recurrent sequence:

Since can be chosen arbitrarily, we can identify their coefficients in the two expressions obtained from the recurrent sequence in question. We thus obtain:

(4)

And also:

(5)

By making an adequate subtraction between and , we can obtain:

Where:

e) Examples.

Using the preceding relations, one can establish several formulas and values of sums. We will cite a few examples below, especially for cases and (case of the equation of the three moments in structural mechanics).

In the case where is a real, its main argument is therefore either 0 (if it is positive) or (if it is negative). We find in these two cases the following formula:

If we note , then:

And we get for (using limited development):

get for :

6. Conclusion

The twin polynomials are well introduced and several formulas are established. Also, we have well defined the kernel matrix and we have shown that it is orthogonal. We have thus proposed a method for solving recurrent sequences of order 3. Finally, other future researches will be based on this article as a fundamental reference.

Acknowledgements

The author is grateful for the corrections and suggestions of the anonymous referee.

References