Applications of some special numbers obtained from a difference equation of degree three

TL;DR

This paper explores applications of special numbers derived from a third-degree difference equation, focusing on their use in coding theory and extending these numbers to generalized quaternions.

Contribution

It introduces generalized Pell-Fibonacci-Lucas numbers from a third-degree difference equation and extends their application to generalized quaternions, highlighting new mathematical connections.

Findings

Derived special numbers from a third-degree difference equation.

Applied these numbers to coding theory.

Extended the numbers to generalized quaternions.

Abstract

In this paper we present applications of some special numbers obtained from a difference equation of degree three, especially in the Coding Theory. As a particular case, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternions.

Equations

$$D_n=a{D}_{n-1}+b{D}_{n-2}+c{D}_{n-3},D_0=x_0,D_1=x_1,D_2=x_2.$$

$$c\left(x\right)=c_0+c_{1}x+c_2{x}^2+...+c_{n-1}{x}^{n-1},$$

$$c^{\prime }\left(x\right)=c_{n-1}+c_{0}x+c_1{x}^2+c_2{x}^3+...+c_{n-2}{x}^{n-1}.$$

$$c^{\prime }\left(x\right)=xc\left(x\right)-c_{n-1}\left(x^n-1\right),$$

$$c\left(x\right)\in Cmod\left(x^n-1\right)\text{ if and only if }xc\left(x\right)\in Cmod\left(x^n-1\right).$$

$$\min \{d_H\left(c_1,c_2\right)/c_1,c_2\in C\}.$$

$$\min \{w_H\left(c_1,c_2\right)/c_1,c_2\in C\}.$$

$$s_1\left(x\right)=xs\left(x\right)\text{mod }f\left(x\right)$$

$$s_2\left(x\right)=x^{2}s\left(x\right)\text{mod }f\left(x\right)....$$

$$s_i\left(x\right)=x^{i}s\left(x\right)\text{mod }f\left(x\right),$$

$$\delta \left(x\right)\left(c{x}^3+b{x}^2+ax-1\right)=$$

$$=c{D}_{l-1}{x}^{l+2}+D_{l+1}{x}^{l+1}+\left(a{D}_1+b{D}_0-D_2\right)x^2+\left(a{D}_0-D_1\right)x-D_0.$$

$$D_n=a{D}_{n-1}+b{D}_{n-2},$$

$$P_n=2P_{n-1}+P_{n-2},n\ge 2,P_0=0;P_1=1,$$

$$Q_n=2Q_{n-1}+Q_{n-2},n\ge 2,Q_0=2;Q_1=2.$$

$$P_n=\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }=\frac{{\alpha }^n-{\beta }^n}{2\sqrt{2}},\left(\forall \right)n\in \mathbb{N}.$$

$$Q_n={\alpha }^n+{\beta }^n,\left(\forall \right)n\in \mathbb{N}.$$

$$A\left(z\right)=\frac{D_0+\left(D_1-a{D}_0\right)z+\left(D_2-a{D}_1-b{D}_0\right)z^2}{1-az-b{z}^2-c{z}^3}.$$

$$A\left(z\right)=D_0+D_{1}z+D_2{z}^2+D_3{z}^3+...+D_n{z}^n+...$$

$$azA\left(z\right)=a{D}_{0}z+a{D}_1{z}^2+a{D}_2{z}^3+...+a{D}_{n-1}{z}^n+...$$

$$b{z}^{2}A\left(z\right)=b{D}_0{z}^2+b{D}_1{z}^3+b{D}_2{z}^4+...+b{D}_{n-2}{z}^n+...$$

$$c{z}^{3}A\left(z\right)=c{D}_0{z}^3+c{D}_1{z}^4+c{D}_2{z}^5+...+c{D}_{n-3}{z}^n+...$$

$$A\left(z\right)\left(1-az-b{z}^2-c{z}^3\right)=D_0+\left(D_1-a{D}_0\right)z+\left(D_2-a{D}_1-b{D}_0\right)z^2.$$

$$A\left(z\right)=\frac{D_0+\left(D_1-a{D}_0\right)z+(D_2-a{D}_1-b{D}_0)z^2}{1-az-b{z}^2-c{z}^3}.$$

$$D_{n+3}+D_{n+2}=2\left(D_{n+2}+D_{n+1}\right)+\left(D_{n+1}+D_n\right).$$

$$b_{n+3}=2b_{n+2}+b_{n+1},n\ge 0,$$

$$A^{n}=\left(\begin{array}[]{lll}\frac{Q_{n}}{2}&0&\sqrt{2}P_{n}\\ 0&\frac{Q_{n}}{2}+\sqrt{2}P_{n}&0\\ \sqrt{2}P_{n}&0&\frac{Q_{n}}{2}\end{array}\right).$$

$$P\left(n\right):A^{n}=\left(\begin{array}[]{lll}\frac{Q_{n}}{2}&0&\sqrt{2}P_{n}\\ 0&\frac{Q_{n}}{2}+\sqrt{2}P_{n}&0\\ \sqrt{2}P_{n}&0&\frac{Q_{n}}{2}\end{array}\right).$$

$$P\left(n+1\right):A^{n+1}=\left(\begin{array}[]{lll}\frac{Q_{n+1}}{2}&0&\sqrt{2}P_{n+1}\\ 0&\frac{Q_{n+1}}{2}+\sqrt{2}P_{n+1}&0\\ \sqrt{2}P_{n+1}&0&\frac{Q_{n+1}}{2}\end{array}\right).$$

$$A^{n+1}=A^{n}A=\left(\begin{array}[]{lll}\frac{Q_{n}}{2}&0&\sqrt{2}P_{n}\\ 0&\frac{Q_{n}}{2}+\sqrt{2}P_{n}&0\\ \sqrt{2}P_{n}&0&\frac{Q_{n}}{2}\end{array}\right)\cdot$$

Full text

Applications of some special numbers obtained from a difference equation of degree three

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Cristina FLAUT and Diana SAVIN

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**Abstract. **In this paper we present applications of special numbers obtained from a difference equation of degree three. One of these applications is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes). In another particular case of these difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were  extended to the generalized quaternion algebras. Using properties of these elements, we can define a set with an interesting algebraic structure, namely an order of a generalized rational quaternion algebra.

Key Words: Cyclic codes; Fibonacci numbers; Pell numbers; Quaternion algebras.

2000 AMS Subject Classification: 11B39, 11R54, 94B05, 94B15.

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1. Introduction

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Let $n$ be an arbitrary positive integer and let $a,b,c,x_{0},x_{1},x_{2}~{}$be arbitrary integers. We consider the following difference equation of degree three

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If we consider $a=b=1,c=0,$ $x_{0}=0,x_{1}=1,x_{2}=1,$ we obtain the Fibonacci numbers and if we take $a=b=1,c=0,x_{0}=2,x_{1}=1,x_{2}=3,$ we get the Lucas numbers. If we consider $a=2,b=1,c=0,x_{0}=0,x_{1}=1,x_{2}=2,$ we obtain the Pell numbers and if we take $a=1,b=0,c=1,x_{0}=0,x_{1}=1,x_{2}=1,$ we find the Fibonacci-Narayana numbers.

Some properties of the above numbers were studied in various paper. In this paper, we will provide properties and applications of other special numbers obtained from the equation $\left(1.1\right),$ as from example in the Coding Theory. Moreover, we extend these numbers to generalized quaternions, obtaining an interesting algebraic structure.

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2. An application in the Coding Theory

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In [Ba, Pr; 09], [St; 06] and [Ko, Oz, Si; 17] were presented some applications of the Fibonacci elements in the Coding Theory. In the following, we will give applications of other special numbers in this domain.

Let $F_{\pi},$ $\pi$ a prime number, be a finite field and let $C\subset F_{\pi}^{n},$ be a linear code. Let $c=\left(c_{0},c_{1},c_{2},...,c_{n-1}\right)\in C$   be a code-word. A linear code $C$ of length $n$ over a finite field $F_{\pi}~{}$is a cyclic code if $c=\left(c_{0},c_{1},c_{2},...,c_{n-1}\right)\in C$ implies that $c^{\prime}=\left(c_{n-1},c_{0},c_{1},c_{2},...,c_{n-2}\right)\in C.$ From here, we have that $C$ is invariant at a single right cyclic shift. It is very useful if each code-word in a cyclic code is represented using polynomials. Therefore to the code-word $c=\left(c_{0},c_{1},c_{2},...,c_{n-1}\right)$ we associate the polynomial

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called the associated code polynomial. It results that the associated code polynomial for $c^{\prime}$ is

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We obtain that

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therefore $c^{\prime}\left(x\right)=xc\left(x\right)~{}$mod $\left(x^{n}-1\right).$ From here, we have that

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A code polynomial $f\left(x\right)$ generates the cyclic code $C$ of length $n$ if and only if $f\left(x\right)\mid\left(x^{n}-1\right).$ The polynomial $f$ is called the generator polynomial for the code $C$. The polynomial $g\left(x\right)=\frac{x^{n}-1}{f\left(x\right)}$ is called the check polynomial of the code  $C.$

For a code $C,$ the Hamming distance, $d,$ between two code-words of the same length $n~{}$is the number of positions at which the corresponding symbols are different. The minimum *Hamming distance *of a code, $d_{H}$ is

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The Hamming weight, $w,$ of a code is the number of symbols which differ from the zero-symbol of the used alphabet. The minimum *Hamming weight *of a code, $w_{H}$ is

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Let $m\left(x\right)$ be a transmitted message, $r\left(x\right)$ a received message and $e\left(x\right)$ the error occurred. The syndrome of the received vector is $s\left(x\right)=r\left(x\right)$ mod$~{}f\left(x\right).$ Supposing that the code $C$ is $t-$errors corrected code, we compute $s_{1}\left(x\right)=xs\left(x\right)$ mod $f\left(x\right)$

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until $w_{H}\left(s_{i}\right)\leq t$. It results the error $e\left(x\right)=x^{n-i}s_{i}~{}$mod $\left(x^{n}-1\right),$ therefore $m\left(x\right)=r\left(x\right)-e\left(x\right).$For  other details regarding cyclic codes, the reader is referred, for example, to [Li, Xi; 04].

**Definition 2.1. **Let  $C$ be a linear code, with parameters $[n,k,d_{H}]$, with $n$ the length of the codewords, $k$ the dimension of the code. If we have $k+d=n+1,$ the code $C$ is called maximum distance separable code, shortly MDS.

Let $\pi$ be a positive prime integer. We consider the degree three sequence given in $\left(1.1\right).$ We denote with $l_{d}\left(\pi\right)$ and $\beta_{d}\left(\pi\right)$ the period, respectively the number of zeros in a single period for this sequence defined on $\mathbb{Z}_{\pi},$where $\mathbb{Z}_{\pi}$ denotes the finite field of integers modulo $\pi,$ having $\pi$ elements.

We consider $\delta\left(x\right)\in\mathbb{Z}_{\pi}[x],\delta\left(x\right)=\overset{l_{d}\left(\pi\right)}{\underset{i=0}{\sum}}D_{i}x^{i},$ the $\mathbf{D}-$polynomial associated to the sequence $\mathbf{D,}$ given by $\left(1.1\right),$ over  $\mathbb{Z}_{\pi},$ where $\mathbf{D}=\{D_{0},D_{1},...D_{n},...\}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

**Theorem 2.2. **With the above notations, let $\delta\left(x\right)\in\mathbb{Z}_{\pi}[x]$ be the $\mathbf{D}-$ polynomial. The following relation is true

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.

Proof. We denote $l_{D}\left(\pi\right)=l.$ Let $D_{n}=aD_{n-1}+bD_{n-2}+cD_{n-3},d_{0}=x_{0},d_{1}=x_{1},d_{2}=x_{2},$be a difference equation of degree three. First of all, we compute

$\delta\left(x\right)\left(cx^{3}+bx^{2}+ax-1\right)=\left(cx^{3}+bx^{2}+ax-1\right)\overset{l-1}{\underset{i=0}{\sum}}D_{i}x^{i}=$

$=cD_{l-1}x^{l+2}+bD_{l-1}x^{l+1}+aD_{l-1}x^{l}-D_{l-1}x^{l-1}+$

$+cD_{l-2}x^{l+1}+bD_{l-2}x^{l}+aD_{l-2}x^{l-1}-D_{l-2}x^{l-2}+$

$+cD_{l-3}x^{l}+bD_{l-3}x^{l-1}+aD_{l-3}x^{l-2}-D_{l-3}x^{l-3}+$

$+cD_{l-4}x^{l-1}+bD_{l-4}x^{l-2}+aD_{l-4}x^{l-3}-D_{l-4}x^{l-4}+...+$

$+cD_{4}x^{7}+bD_{4}x^{6}+aD_{4}x^{5}-D_{4}x^{4}+$

$+cD_{3}x^{6}+bD_{3}x^{5}+aD_{3}x^{4}-D_{3}x^{3}+$

$+cD_{2}x^{5}+bD_{2}x^{4}+aD_{2}x^{3}-D_{2}x^{2}+$

$+cD_{1}x^{4}+bD_{1}x^{3}+aD_{1}x^{2}-D_{1}x+$

$+cD_{0}x^{3}+bD_{0}x^{2}+aD_{0}x-D_{0}.$

Using relation $\left(1.1\right)$, we obtain

$\delta\left(x\right)\left(cx^{3}+bx^{2}+ax-1\right)=$

$=cD_{l-1}x^{l+2}+bD_{l-1}x^{l+1}+aD_{l-1}x^{l}+$

$+cD_{l-2}x^{l+1}+bD_{l-2}x^{l}+cD_{l-3}x^{l}+$

$+aD_{1}x^{2}-D_{1}x-D_{2}x^{2}+bD_{0}x^{2}+aD_{0}x-D_{0}=$

$=cD_{l-1}x^{l+2}+D_{l+1}x^{l+1}+\left(aD_{1}+bD_{0}-D_{2}\right)x^{2}+\left(aD_{0}-D_{1}\right)x-D_{0}.\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

Remark 2.3. For $D_{0}=0,D\,_{1}=1,c=0,b=1,$ since $D_{2}=aD_{1}$ $D_{l+1}=D_{1}~{}$mod $\pi,$ we have $\delta\left(x\right)\left(cx^{3}+bx^{2}+ax-1\right)=\delta\left(x\right)\left(x^{2}+ax-1\right)=$

$=D_{l+1}x^{l+1}+\left(aD_{1}-D_{2}\right)x^{2}-x=x^{l+1}-x.~{}$Therefore, in this situation, we get $\delta\left(x\right)\left(x^{2}+ax-1\right)=x^{l+1}-x.\vskip 6.0pt plus 2.0pt minus 2.0pt$

We consider the following difference equation

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with $D_{0}=0,D\,_{1}=1.\vskip 6.0pt plus 2.0pt minus 2.0pt$

**Theorem 2.4. **With the above notations, let $\delta\left(x\right)\in\mathbb{Z}_{\pi}[x]$ be the $\mathbf{D}-$polynomial associated to the sequence $\left(2.1\right),~{}\delta\left(x\right)=x\delta^{\prime}\left(x\right)$. The cyclic code  $C=<\delta^{\prime}\left(x\right)>,$ generated by $\delta^{\prime}\left(x\right),$ has dimension $2$ and minimum Hamming distance $d=l_{d}\left(\pi\right)-\beta_{d}\left(\pi\right).\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof. We denote $l_{d}\left(\pi\right)=l.$ Using Theorem 2.2 and Remark 2.3, for $D_{0}=0,D\,_{1}=1,c=0,b=1,$we obtain that

$\delta\left(x\right)\left(x^{2}+2x-1\right)=x^{l+1}-x=x\left(x^{l}-1\right).$ It is clear that $\delta^{\prime}\left(x\right)\mid\left(x^{l}-1\right),$ $x\mid\delta\left(x\right)$ and $x\nmid\left(x^{l}-1\right).$ From here, we obtain that gcd$\left(\delta\left(x\right),x^{l}-1\right)=\frac{x^{l}-1}{x^{2}+ax-1}=\frac{\delta\left(x\right)}{x}=\delta^{\prime}\left(x\right).$ We consider the cyclic code of length $l,~{}$generated by $\delta^{\prime}\left(x\right),~{}C=<\delta^{\prime}\left(x\right)>.$ This code has dimension $l-~{}$deg$\left(\delta^{\prime}\left(x\right)\right)=2$, therefore has $\pi^{2}$ elements. From above, $\delta^{\prime}\left(x\right)=\frac{\delta\left(x\right)}{x},$ therefore the polynomials $\delta^{\prime}\left(x\right)$ and $\delta\left(x\right)$ have the same weight. Since the number of the zero elements in a single period for a $D-$sequence is $\beta_{d}\left(\pi\right),$ we have that the minimum Hamming weight is $w\left(\delta^{\prime}\left(x\right)\right)=l-\beta_{d}\left(\pi\right).$ To prove that the minimum Hamming distance is $l-\beta_{d}\left(\pi\right),$ we consider the polynomials $\delta^{\prime}\left(x\right)$ and $\delta\left(x\right)=x\delta^{\prime}\left(x\right)\in C.$   The polynomial $\delta^{\prime}\left(x\right)$ has $\beta_{d}\left(\pi\right)~{}$zero elements on the positions translated with one to the right, compared with the position of the zeros in the given period  and the polynomial $x\delta^{\prime}\left(x\right)$ has $\beta_{d}\left(\pi\right)~{}$ zero elements on the positions given by the period. If we consider another polynomial $c\left(x\right)\in$ $C$ with the number of zero elements more than $\beta_{d}\left(\pi\right),$ supposing for example $\beta_{d}\left(\pi\right)+1,$ we have that $\delta^{\prime}\left(x\right),x\delta^{\prime}\left(x\right),c\left(x\right)$ are linearly independent elements, therefore the minimum Hamming distance is $l-\beta_{d}\left(\pi\right).\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

The above Theorem generalized Theorems 2.2, 2.4, 2.6 from [Ko, Oz, Si; 17] to $\mathbf{D}-$polynomials. Using ideas from the above mentioned paper, where were presented cyclic codes obtained from Fibonacci polynomials, in the following, for particular cases, we will present some properties of cyclic codes obtained from $D-$polynomials.

Let $\pi$ be a positive prime integer. We denote with $l_{fib}\left(\pi\right)$ and $\beta_{fib}\left(\pi\right)$ the period, respectively the number of zeros in a single period of the Fibonacci sequence considered in $\mathbb{Z}_{\pi}$ and with $l_{pell}\left(\pi\right)$ and $\beta_{fib}\left(\pi\right)$  the period, respectively the number of zeros in a single period of of the Pell sequence considered in $\mathbb{Z}_{\pi}.\vskip 6.0pt plus 2.0pt minus 2.0pt\vskip 6.0pt plus 2.0pt minus 2.0pt$

**Example 2.5.  **Let $D_{0}=0,D\,_{1}=1,c=0,b=1,$

i) For $a=1,$ we have the Fibonacci numbers. For $\pi=3,$ we have $l_{fib}\left(\pi\right)=8,\beta_{fib}\left(\pi\right)=2$, for $\pi=5,$ we get $l_{fib}\left(\pi\right)=20,\beta_{fib}\left(\pi\right)=4,$ for $\pi=7,$ we have $l_{fib}\left(\pi\right)=16,\beta_{fib}\left(\pi\right)=2,$ for $\pi=11,$ we obtain $l_{fib}\left(\pi\right)=10,\beta_{fib}\left(\pi\right)=1,$ for $\pi=13,$ we have $l_{fib}\left(\pi\right)=28,\beta_{fib}\left(\pi\right)=4.$ (Also you can see [Ko, Oz, Si; 17])

ii) For $a=2,$ we get the Pell numbers. For $\pi=3,$ we have $l_{pell}\left(\pi\right)=8,\beta_{pell}\left(\pi\right)=2.$ If $\pi=5,$ we obtain $l_{pell}\left(\pi\right)=12,\beta_{pell}\left(\pi\right)=4$ and if $\pi=7,$ we have $l_{pell}\left(\pi\right)=6,\beta_{pell}\left(\pi\right)=1.$ For $\pi=11,$ we have $l_{pell}\left(\pi\right)=24,\beta_{pell}\left(\pi\right)=1$ and for $\pi=13,$ we get $l_{pell}\left(\pi\right)=28,\beta_{pell}\left(\pi\right)=4.$

iii) Case  $a=3.$ For example, if $\pi=11,$ we have $l_{3}\left(\pi\right)=8,\beta_{3}\left(\pi\right)=2.$

iv) Case  $a=4.$ For example, if $\pi=11,$ we get $l_{4}\left(\pi\right)=10,\beta_{4}\left(\pi\right)=1.$

v) Case  $a=5.$ For example, if $\pi=7,$ we obtain $l_{5}\left(\pi\right)=6,\beta_{5}\left(\pi\right)=1.$

vi) Case  $a=6.$** **For example, if $\pi=7,$ we have $l_{7}\left(\pi\right)=4,\beta_{7}\left(\pi\right)=1~{}$and if $\pi=13,$ we get $l_{13}\left(\pi\right)=6,\beta_{13}\left(\pi\right)=1.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Example 2.6. i) We consider the Pell numbers for $\pi=7.$ We have $l_{pell}\left(\pi\right)=6,\beta_{pell}\left(\pi\right)=1.~{}$The corresponding Pell sequence modulo $7$ is

$0,1,2,5,5,1,$ and we obtain the Pell-polynomial, $p\left(x\right)=x+2x^{2}+5x^{3}+5x^{4}+x^{5}.$ The code $C,$ generated by the polynomial $p^{\prime}\left(x\right),$ has minimum Hamming distance $d=5,$ and it is on the type $[6,2,5].$Since $k+d=7=n+1,$ the code $C$ is an MDS code.

ii) We consider $a=5$ and $\pi=7.$ We have $l_{5}\left(\pi\right)=6,\beta_{5}\left(\pi\right)=1.$The corresponding $\mathbf{D}-$sequence modulo $7$ is $0,1,5,5,2,1$ and we obtain the $\mathbf{D}$-polynomial $\delta\left(x\right)=x+5x^{2}+5x^{3}+2x^{4}+x^{5}.$ The code $C,$ generated by the polynomial $\delta^{\prime}\left(x\right),$ has Hamming distance $d=5~{}$and it is on the type $[6,2,5],$ being an MDS code.

iii) We consider $a=6$ and $\pi=13.$ We have $l_{6}\left(\pi\right)=6,\beta_{6}\left(\pi\right)=1.$The corresponding $\mathbf{D}-$sequence modulo $13$ is $0,1,6,11,7,1$ and we obtain the $\mathbf{D}$-polynomial $\delta\left(x\right)=x+6x^{2}+11x^{3}+7x^{4}+x^{5}.$ The code $C,$ generated by the polynomial $\delta^{\prime}\left(x\right),$ has Hamming distance $d=5,$ and it is on the type $[6,2,5],$ being an MDS code.

Example 2.7. i) We consider $a=3$ and $\pi=11.$ We have $l_{3}\left(\pi\right)=8,\beta_{3}\left(\pi\right)=2.$The corresponding $\mathbf{D}-$sequence modulo $11$ is $0,1,3,10,0,10,8,1$ and we obtain the $\mathbf{D}$-polynomial $\delta\left(x\right)=x+3x^{2}+10x^{3}+10x^{5}+8x^{6}+x^{7}.$ The code $C,$ generated by the polynomial $\delta^{\prime}\left(x\right),$ has Hamming distance $d=6,$ and it is on the type $[8,2,6].~{}$Since $k+d=7=n+1,$ this code is not an MDS code.

ii) For $a=8$ and $\pi=13,$ we have $l_{8}\left(\pi\right)=12,\beta_{8}\left(\pi\right)=4.$The corresponding $\mathbf{D}-$sequence modulo $13$ is $0,1,8,0,8,12,0,12,5,0,5,1$ and we obtain the $\mathbf{D}$-polynomial $\delta\left(x\right)=x+8x^{2}+8x^{4}+12x^{5}+12x^{7}+5x^{8}+5x^{10}+x^{11}.$ The code $C,$ generated by the polynomial $\delta^{\prime}\left(x\right),$ has Hamming distance $d=8,$ is on the type $[8,2,8]$ and it is not an MDS code.

Example 2.8. We consider for $\pi=7$, the Pell-polynomial, $p\left(x\right)=x^{5}+5x^{4}+5x^{3}+$ $2x^{2}+x~{}$and the code $C$ generated by the polynomial $p^{\prime}\left(x\right)=x^{4}+5x^{3}+5x^{2}+2x+1.~{}$This code is $2-$errors correcting code. We suppose that the code-word sent was $c=\left(0,3,6,1,1,3\right)$ and we received the code-word $r=\left(0,5,6,1,1,6\right).$ The corresponding polynomial for the code-word $c$ is $3x+6x^{2}+x^{3}+x^{4}+3x^{5}.$ We compute the syndrome polynomial. We get

$6x^{5}+x^{4}+x^{3}+6x^{2}+5x=$

$=\left(x^{4}+5x^{3}+5x^{2}+2x+1\right)\left(6x+6\right)+4x^{3}+6x^{2}+x+1,$ therefore the syndrome is $s\left(x\right)=(4x^{3}+6x^{2}+x+1)$ mod $\left(x^{4}+5x^{3}+5x^{2}+2x+\right).$ We have

$s_{1}\left(x\right)=xs\left(x\right)$ mod $\left(x^{4}+5x^{3}+5x^{2}+2x+1\right)=$

$=(4x^{4}+6x^{3}+x^{2}+x)$ mod $\left(x^{4}+5x^{3}+5x^{2}+2x+1\right)=$

$=4\left(x^{4}+5x^{3}+5x^{2}+2x+1\right)+2x^{2}+3~{}$mod $\left(x^{4}+5x^{3}+5x^{2}+2x+1\right).$

Therefore $s_{1}\left(x\right)=2x^{2}+3$ mod p^{\prime}.\,\It results that $i=1$ and the error polynomial is $e\left(x\right)=x^{6-1}s_{1}\left(x\right)$ mod $\left(x^{6}-1\right).$ Since $2x^{7}+3x^{5}=2x\left(x^{6}-1\right)+3x^{5}+2x,$ we obtain that $(2x^{7}+3x^{5})~{}$mod $\left(x^{6}-1\right)=3x^{5}+2x.$ Therefore, the submitted word was $c=r-e=$ $\left(0,5,6,1,1,6\right)-\left(0,2,0,0,0,3\right)=\left(0,3,6,1,1,3\right),$that means we recover the submitted word.

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3. Some applications of **generalized Pell-Fibonacci-Lucas elements **

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In relation $\left(1.1\right),$ if we consider $a=2,b=1,c=0,x_{0}=0,x_{1}=1,x_{2}=2,$ we obtain the Pell numbers and if we take $a=2,b=1,c=0,x_{0}=2,x_{1}=2,x_{2}=6,$ we obtain the Pell-Lucas numbers. Let $\left(P_{n}\right)_{n\geq 0}$ be the Pell sequence

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and $\left(Q_{n}\right)_{n\geq 0}$ be the Pell-Lucas sequence

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We consider the numbers $\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}.$ The following formulae are well known:

Binet’s formula for Pell sequence.

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Binet’s formula for Pell-Lucas sequence.

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Let $A$ be the generating function for the sequence $\left(D_{n}\right)_{n\geq 0}$,  given by the relation $\left(1,1\right)$ $A\left(z\right)=\sum\limits_{n\geq 0}D_{n}z^{n}.$ In the following, we determine this function.

Proposition 3.1. We have:

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Proof.

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Adding the equalities (3.2), (3.3) and (3.4) member by member, we obtain:

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Therefore, we get

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$\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

For $a=c=1$ and $b=3$ we obtain the sequence $\left(D_{n}\right)_{n\geq 0},$ $D_{0}=0,$ $D_{1}=D_{2}=1,$ $D_{n+3}=D_{n+2}+3D_{n+1}+D_{n},\;n\geq 0.$ This equality is equivalent with

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If we take the sequence $\left(b_{n}\right)_{n\geq 0},$ $b_{n+1}=D_{n+1}+D_{n},\;n\geq 0,$ the last equality becomes

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where $b_{1}=1$ and $b_{2}=2.$ Moreover, if we consider $b_{0}=0,$ it results that the sequence $\left(b_{n}\right)_{n\geq 0}$ is in fact the sequence of Pell numbers $\left(P_{n}\right)_{n\geq 0}.\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proposition 3.2. Let $\left(P_{n}\right)_{n\geq 0}$ be the sequence of Pell numbers and $\left(Q_{n}\right)_{n\geq 0}$ be the sequence of Pell-Lucas numbers. Let $A$ *be the following matrix

A=\left(\begin{array}[]{lll}\frac{Q_{1}}{2}&0&\sqrt{2}P_{1}\\ 0&\frac{Q_{1}}{2}+\sqrt{2}P_{1}&0\\ \sqrt{2}P_{1}&0&\frac{Q_{1}}{2}\end{array}\right).* Then, we have:

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Proof. We prove by induction after n$$\in$$\mathbb{N}^{\ast} the following statement

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We remark that $P\left(1\right)$ is true.

We suppose that $P\left(n\right)$ is true and we prove that $P\left(n+1\right)$ is true.

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Using Binet’s formulas for Pell-Lucas sequence and Pell-Lucas sequence, we obtain:

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Therefore, $P\left(n+1\right)$ is true.$\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

In the paper [Fl, Sa; 15], we introduced the generalized Fibonacci-Lucas numbers and the generalized Fibonacci-Lucas quaternions and we obtained many properties of these elements ( also you can see [Sa; 17]). In a similar way, we introduce here the generalized Pell-Fibonacci-Lucas numbers and the generalized Pell-Fibonacci-Lucas quaternions.

First, we will give some identities involving Pell numbers and Pell-Lucas numbers.

**Proposition 3.3. ** Let $\left(P_{n}\right)_{n\geq 0}$ be the sequence of Pell numbers and $\left(Q_{n}\right)_{n\geq 0}$ be the sequence of Pell-Lucas numbers. Then, we have:

i)

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ii)

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iii)

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iv)

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Proof. We will use Binet’s formulas for Pell numbers and Pell-Lucas numbers.

i)

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ii)

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iii)

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iv)

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$\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

Let $n$ be an arbitrary positive integer and $p,q$ be two arbitrary integers. We introduce the following numbers $\left(r_{n}\right)_{n\geq 1},$

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We call the numbers $r_{n}$ $(n\geq 1)$ the generalized Pell-Fibonacci-Lucas numbers.

To avoid confusions, in the followings we will use instead of $r_{n}$ the notation $r_{n}^{p,q}.$

In the following, we consider $\alpha,\beta\in\mathbb{Q}\setminus\{0\}$. Let $\mathbb{H}\left(\alpha,\beta\right)$ be the generalized quaternion algebra with basis $\{1,e_{1},e_{2},e_{3}\}.$ This algebra is a rational algebra with the multiplication given in the following table

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We consider \alpha,\beta$$\in$$\mathbb{Q}^{\ast} and $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right)$ the generalized quaternion algebra over the rational field. We define the $n$-th *generalized Pell- Fibonacci-Lucas quaternion *to be the element

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**Remark 3.4. ** Let $n$ be an arbitrary positive integer and $p,q$ be two arbitrary integers. Let $\left(r_{n}^{p,q}\right)_{n\geq 1}$ be the generalized Pell-Fibonacci-Lucas numbers and $\left(R_{n}^{p,q}\right)_{n\geq 1}$ be the generalized Pell-Fibonacci-Lucas quaternion elements. Then we have the following relation

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Proof. ” $\Leftarrow$” It is trivial.

” $\Rightarrow$” Using the fact that $\left\{1,i,j,k\right\}$ is a basis in $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right),$ we obtain that $r_{n}^{p,q}=r_{n+1}^{p,q}=r_{n+2}^{p,q}=r_{n+3}^{p,q}=0.$ This implies that $r_{n-1}^{p,q}=0$ …, $r_{2}^{p,q}=0$, $r_{1}^{p,q}=0.$ Since $r_{1}^{p,q}=pP_{0}+qQ_{1}=2q,$ it results $q=0.$ From $r_{2}^{p,q}=0,$ we obtain $p=0.\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

**Remark 3.5. ** Let $n$ be an arbitrary positive integer and $p,q$ be two arbitrary integers. Let $\left(r_{n}^{p,q}\right)_{n\geq 1}$ be the generalized Pell- Fibonacci-Lucas numbers. The following relation is true

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Proof.

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$\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

**Definition 3.6. ** A subring $O\subseteq\mathbb{H}(\alpha,\beta)$ is an order in $\mathbb{H}(\alpha,\beta)$ if $O$ is a finitely generated $\mathbb{Z}$-submodule of $\mathbb{H}(\alpha,\beta)$ ( see [Vo; 15]).

**Proposition 3.7. ** Let $O$ be the set

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This set is an order of the quaternion algebra $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right).$

Proof. We remark that [math]\in$$O (according to Remark 3.4). We will show that $O$ is a $\mathbb{Z}-$ submodule of $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right).$ Indeed, if $n,m\in\mathbb{N}^{\ast},$ $a,b,p,q,p^{{}^{\prime}},q^{{}^{\prime}}\in\mathbb{Z},$ it results immediately that

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which implies the following relation

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So, we obtain that $O$ is a free $\mathbb{Z}-$ submodule of rank $4$ for the quaternion algebra $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right).$

To obtain an order, we will prove that $O$ is a subring of $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right).$ Let $m,n$ be two integers, $n<m.$ We calculate

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Using Proposition 3.3 (i, ii, iii, iv), Remark 3.5 and the equality (3.5), it results:

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We get that 8r_{n}^{p,q}\cdot 8r_{m}^{p^{{}^{\prime}},q^{{}^{\prime}}}$$\in$$O. From here we obtain that $O$ is an order of the quaternion algebra $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right).\Box\vskip 3.0pt plus 1.0pt minus 1.0pt$

Conclusions. In this paper, we provided applications of  several cases of difference equation of degree three. One of these applications is in the Coding Theory and we used the obtained numbers to built, in some special situations, cyclic codes with good properties. We have defined the generalized Pell-Fibonacci-Lucas quaternion elements and we proved that, using these elements, we can get a set which is * *an order of the generalized quaternion algebra $\mathbb{H}_{\mathbb{Q}}\left(\alpha,\beta\right).$

Using these approaches, the above obtained results can constitute the start for a further research in which we intend to study properties and applications of other difference equations.

Acknowledgments. The authors thank referees for their suggestions and remarks which helped us to improve this paper.

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**References

**

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[Ba, Pr; 09] M. Basu, B. Prasad, The generalized relations among the code elements for Fibonacci coding theory, Chaos, Solitons and Fractals, 41(2009), 2517-2525.

[Fl, Sa; 15] C. Flaut, D. Savin, Quaternion Algebras and Generalized Fibonacci-Lucas Quaternions, Adv. Appl. Clifford Algebras, 25(4)(2015), 853-862.

[Fl, Sh;13] C. Flaut, V. Shpakivskyi, On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions, Adv. Appl. Clifford Algebras, 23(3)(2013), 673–688.

[Fib.] http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html

[Ko, Oz, Si; 17] M. E. Koroglu, I. Ozbek, I. Siap, Optimal Codes from Fibonacci Polynomials and Secret Sharing Schemes, Arab. J. Math, 2017, 1-12, DOI 10.1007/s40065-017-0171-7, https://link.springer.com/content/pdf/10.1007%2Fs40065-017-0171-7.pdf

[Li, Xi; 04] S. Ling, C. Xing, Coding Theory-A first course, Cambridge University Press, 2004.

[Sa; 17] D. Savin, About Special Elements in Quaternion Algebras Over Finite Fields, Adv. Appl. Clifford Algebras, 27(2)(2017), 1801-1813.

[St; 06] A.P. Stakhov, Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory, Chaos, Solitons and Fractals, 30(2006), 56-66.

[Vo; 15] J.Voight, *The arithmetic of quaternion algebras, *2015, available on the author’s website: http://www.math.dartmouth.edu/jvoight/crmquat/book/quatmodforms-041310.

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Cristina FLAUT

Faculty of Mathematics and Computer Science, Ovidius University,

Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA

http://cristinaflaut.wikispaces.com/; http://www.univ-ovidius.ro/math/

e-mail: cflaut@univ-ovidius.ro; cristina_flaut@yahoo.com

Diana SAVIN

Faculty of Mathematics and Computer Science,

Ovidius University,

Bd. Mamaia 124, 900527, CONSTANTA, ROMANIA

http://www.univ-ovidius.ro/math/

e-mail:  savin.diana@univ-ovidius.ro,  dianet72@yahoo.com