A Note on the Generalized k-Order F&L Hybrinomials

1. Introduction

F&L numbers, which have the most interesting properties and relationships in the world of the mathematics, have been found interesting by many authors. Fibonacci numbers are defined by the recurrence relation of $F_n=F_{n-1}+F_{n-2}$ , with initial conditions $F_0=0 , F_1=1$ for $n\ge 2$. With a similar definition, the recurrence relation of Lucas numbers can be given as $L_n=L_{n-1}+L_{n-2}$for $n\ge 2$ with the initial condition of $L_0=0 , L_1=1$. (for more details see [1,2,3]). One of the most important and popular features of the Fibonacci sequence is its matrix representation so-called Fibonacci $Q$-matrix in [4] as follows:

$$Q=\left[\begin{array}{cc}{F}_2& F_1\\ F_1& F_0\end{array}\right]=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$$

Also, $nth$-power of the Fibonacci $Q$-matrix is shown in [5] by

$$Q^n=\left[\begin{array}{cc}{F}_{n+1}& F_n\\ F_n& F_{n-1}\end{array}\right].$$

Fibonacci polynomials were first studied in 1883 by Belgian mathematician E. Charles Catalan and German mathematician E. Jacobsthal. The Fibonacci polynomials studied by Catalan were later developed by M. N. Swamy in 1966. In addition, a new Fibonacci type polynomial was added to the literature by P. F. Bryd in 1963. The polynomial defined by P. F. Bryd is today called the Pell polynomial. The polynomial accepted as the Fibonacci polynomial is the polynomial defined by Catalan. Later, all these different definitions were named as F&L type polynomials.

The recurrence relation of Fibonacci polynomials defined by Catalan is defined as follows:

$$f_n\left(x\right)=x{f}_{n-1}\left(x\right)+f_{n-2}\left(x\right)$$

where $f_1\left(x\right)=1,f_2\left(x\right)=x$ for $n\ge 3$. It can be easily seen that $f_n\left(1\right)=F_n$ , where $F_n$ is the $nth$ Fibonacci number.

Generalized k-order F&L numbers that were introduced in [6] are defined by the following recurrence relation for integers $d_i$;

$$V_n^{\left(k\right)}=d_1{V}_{n-1}^{\left(k\right)}+d_2{V}_{n-2}^{\left(k\right)}+d_3{V}_{n-3}^{\left(k\right)}+\dots +d_k{V}_{n-k}^{\left(k\right)} (n>k\ge 2)$$

with initial conditions $V_1^{\left(k\right)}=V_2^{\left(k\right)}=V_3^{\left(k\right)}=\dots =V_{k-2}^{\left(k\right)}=0,V_{k-1}^{\left(k\right)}=q,V_k^{\left(k\right)}=d_1$.

The concept of the hybrid numbers and polynomials is a field gaining increasing prominence in mathematics, physics and engineering. Hybrid numbers are a generalization of complex, hyperbolic and dual numbers. Until today, many authors have studied on these numbers. Ulrych in 2005 in [22] explored the potential of hyperbolic numbers in relativistic quantum physics, demonstrating their use as a generalization of complex numbers within this framework. This work highlights the potential for hybrid numbers to offer new perspectives and tools for understanding quantum phenomena. Also, Branicky in 1998 in [23] provided a significant contribution to the analysis of hybrid systems by introducing multiple Lyapunov functions as a tool for stability analysis, expanding the available techniques for understanding these complex systems. The non-commutative number system including these three number systems were defined in [7]. The set of hybrid numbers is defined as

$$K=\left\{a+bi+c\epsilon +dh:a,b,c,d\in \mathbb{R},i^2=-1,{\epsilon }^2=0,h^2=1,ih=-hi=\epsilon +1\right\}$$

(1)

Let $P=a_1+b_{1}i+c_1\epsilon +d_{1}h$ and $Q=a_2+b_{2}i+c_2\epsilon +d_{2}h$ be any two hybrid numbers. The operation of equality, addition, subtraction and multiplication by scalar are defined as follows:

The multiplication of hybrid numbers is defined in the following table using Eq. (1) as:

Horadam hybrid numbers were introduced in [8] for the first time. Later, in [9], Fibonacci hybrid numbers were studied and some important properties about the hybrid numbers were presented. (for a more details of these numbers, see also [10,11,12]). Besides these, Jacobsthal, Jacobsthal-Lucas hybrid numbers and the Pell, Pell-Lucas hybrid numbers were introduced in [13,14] by using the terms of these sequences. More information regarding these sequences can be found in [15,16,17,18]. A new generalization based on these studies was defined in [19]. Also, some algebraic properties regarding F&L hybrid numbers were given in it. In [6], these studies were generalized and generalized k-order F&L hybrid numbers were defined. Furthermore, in the same study matrix representations of generalized k-order F&L hybrid numbers were described by using Table 1. In their work titled "Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation", Akhtamova in [21] investigated recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation.

In this study, we define generalized k-order F&L polynomials, that is a generalization of the generalized k-order F&L numbers. We also obtain similar generalization for hybrid numbers as well. Finally, we define the matrix representations for generalized k-order F&L hybrinomials and give some properties about the matrix representations.

2. Generalized k-Order F&L Polynomials

In this section, we define generalized k-order F&L polynomials. Then, we give some special cases of generalized k-order F&L polynomials such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and many special polynomials.

Definition 2.1. For $n>0$, the $nth$ generalized k-order F&L polynomials are defined by the following recurrence relations

$$\begin{gathered} V_n^{\left(k\right)}\left(x\right)=x^{k-1}{d}_1{V}_{n-1}^{\left(k\right)}\left(x\right)+x^{k-2}{d}_2{V}_{n-2}^{\left(k\right)}\left(x\right)+x^{k-3}{d}_3{V}_{n-3}^{\left(k\right)}\left(x\right)+\dots +x{d}_{k-1}{V}_{n-k-1}^{\left(k\right)}\left(x\right) \\ +d_k{V}_{n-k}^{\left(k\right)}\left(x\right) \end{gathered}$$

(2)

with initial conditions for $n>k\ge 2$

$$V_1^{\left(k\right)}\left(x\right)=V_2^{\left(k\right)}\left(x\right)=V_3^{\left(k\right)}\left(x\right)=\cdots =V_{k-2}^{\left(k\right)}\left(x\right)=0,V_{k-1}^{\left(k\right)}\left(x\right)=q,V_k^{\left(k\right)}\left(x\right)=d_1{x}^{k-1}.$$

Some special cases of generalized $k$-order F&L polynomials are in Table 2 as follows:

1. For $k=2$; we get the following table:

2. For $k=3$ and $d_1=d_2=d_3=1,q=1$; we get Tribonacci polynomials. The recurrence relation of Tribonacci polynomials is

$$T_n\left(x\right)=x^2{T}_{n-1}\left(x\right)+x{T}_{n-2}\left(x\right)+T_{n-3}\left(x\right)$$

with the boundary conditions $T_1\left(x\right)=0,T_2\left(x\right)=1,T_3\left(x\right)=x^2$.

3. For $d_1=d_2=d_3=.\cdots =d_k=1$ and $q=1$, we get $k$-order Fibonacci polynomials.

4. For $d_1=d_2=d_3=\cdots =d_k=1$ and $q=2$, we get $k$-order Lucas polynomials.

5. For $d_1=2,d_2=d_3=\cdots =d_k=1$ and $q=1$, we get $k$-order Pell polynomials.

6. For $d_1=2,d_2=d_3=\cdots =d_k=1$ and $q=2$, we get $k$-order Pell-Lucas polynomials.

7. For $d_1=1,d_2=2,d_3\cdots =d_k=1$ and $q=1$, we get $k$-order Jacobsthal polynomials.

8. For $d_1=1,d_2=2,d_3=\cdots =d_k=1$ and $q=2$, we get $k$-order Jacobsthal-Lucas polynomials.

It is easy to see that generalized $k$-order F&L numbers are obtained by selecting $x=1$ specifically as shown in [4].

3. Generalized k-Order Fibonacci and Lucas Hybrinomials

In this section, we define hybrinomials using generalized $k$-order F&L polynomials. We describe the recurrence relation of hybrinomials, generating functions and give some special results.

Definition 3.1. The $nth$ generalized $k$-order F&L hybrinomials $H{V}_n^{\left(k\right)}\left(x\right)$ are defined a

$$H{V}_n^{\left(k\right)}\left(x\right)=V_n^{\left(k\right)}\left(x\right)+i{V}_{n+1}^{\left(k\right)}\left(x\right)+\epsilon {V_{n+2}^{\left(k\right)}\left(x\right)V}_{n+2}^{\left(k\right)}\left(x\right)+h{V}_{n+3}^{\left(k\right)}\left(x\right)$$

(3)

where $V_n^{\left(k\right)}$is $nth$ generalized $k$-order F&L polynomials.

Some special cases of the generalized $k$-order F&L hybrinomials are as follows:

1. For $k=2$, we obtain some special hybrinomials using (2) in Table 3 as follows:

2. For $k=3$ and $d_1=d_2=d_3=1,q=1$, we get Tribonacci hybrinomials.

Definition 3.2. For every $x\in \mathbb{R}$ , the conjugate of $\overline{H{V}_n^{\left(k\right)}\left(x\right)}$ is defined by

$\overline{H{V}_n^{\left(k\right)}\left(x\right)}=V_n^{\left(k\right)}\left(x\right)-i{V}_{n+1}^{\left(k\right)}\left(x\right)-\epsilon V_{n+2}^{\left(k\right)}\left(x\right)-h{V}_{n+3}^{\left(k\right)}\left(x\right)$.

Theorem 3.1. For every $x\in \mathbb{R}$ , we have the following properties:

i.

$H{V}_n^{\left(k\right)}\left(x\right)+\overline{H{V}_n^{\left(k\right)}\left(x\right)}=2V_n^{\left(k\right)}\left(x\right)$ where $V_n^{\left(k\right)}\left(x\right)$ is the $nth$ generalized $k$-order F&L polynomials.

ii.

$H{V}_n^{\left(k\right)}\left(x\right)\overline{H{V}_n^{\left(k\right)}\left(x\right)}={\left(V_n^{\left(k\right)}\left(x\right)\right)}^2+{\left(V_{n+1}^{\left(k\right)}\left(x\right)\right)}^2-{\left(V_{n+3}^{\left(k\right)}\left(x\right)\right)}^2-2V_{n+1}^{\left(k\right)}\left(x\right)V_{n+2}^{\left(k\right)}\left(x\right)$

iii.

${\left(H{V}_n^{\left(k\right)}\left(x\right)\right)}^2+H{V}_n^{\left(k\right)}\left(x\right)\overline{H{V}_n^{\left(k\right)}\left(x\right)}=2V_n^{\left(k\right)}\left(x\right)\left(V_n^{\left(k\right)}\left(x\right)+H{V}_n^{\left(k\right)}\left(x\right)\right)$where $k\ge 2$ and $n>0.$

Proof.

i.

The proof is easily seen using the definitions of $H{V}_n^{\left(k\right)}\left(x\right)$ and $\overline{H{V}_n^{\left(k\right)}\left(x\right)}$.

ii.

Using $H{V}_n^{\left(k\right)}\left(x\right)$, $\overline{H{V}_n^{\left(k\right)}\left(x\right)}$ and the multiplication of hybrid numbers, we obtain as follows:

$$H{V}_n^{\left(k\right)}\left(x\right)\overline{H{V}_n^{\left(k\right)}\left(x\right)}={\left(V_n^{\left(k\right)}\left(x\right)\right)}^2+{\left(V_{n+1}^{\left(k\right)}\left(x\right)\right)}^2-2V_{n+1}^{\left(k\right)}\left(x\right){\left(V_{n+2}^{\left(k\right)}\left(x\right)\right)}^2-{\left(V_{n+3}^{\left(k\right)}\left(x\right)\right)}^2.$$

iii.

First, we obtain ${\left(H{V}_n^{\left(k\right)}\left(x\right)\right)}^2$ as follows:

$$\begin{gathered} {\left(H{V}_n^{\left(k\right)}\left(x\right)\right)}^2=H{V}_n^{\left(k\right)}\left(x\right)H{V}_n^{\left(k\right)}\left(x\right) \\ ={\left(V_n^{\left(k\right)}\left(x\right)\right)}^2-{\left(V_{n+1}^{\left(k\right)}\left(x\right)\right)}^2+{\left(V_{n+3}^{\left(k\right)}\left(x\right)\right)}^2+2i{V}_n^{\left(k\right)}\left(x\right)V_{n+1}^{\left(k\right)}\left(x\right) \\ +2\epsilon V_n^{\left(k\right)}\left(x\right)V_{n+2}^{\left(k\right)}\left(x\right)+2h{V}_n^{\left(k\right)}\left(x\right)V_{n+3}^{\left(k\right)}\left(x\right)+2V_{n+1}^{\left(k\right)}\left(x\right)V_{n+2}^{\left(k\right)}\left(x\right). \end{gathered}$$

Then, we use (ii). From (ii), we have

$$H{V}_n^{\left(k\right)}\left(x\right)\overline{H{V}_n^{\left(k\right)}\left(x\right)}={\left(V_n^{\left(k\right)}\left(x\right)\right)}^2+{\left(V_{n+1}^{\left(k\right)}\left(x\right)\right)}^2-{\left(V_{n+3}^{\left(k\right)}\left(x\right)\right)}^2-2V_{n+1}^{\left(k\right)}\left(x\right)V_{n+2}^{\left(k\right)}\left(x\right).$$

This we obtain as follows:

$${\left(H{V}_n^{\left(k\right)}\left(x\right)\right)}^2+H{V}_n^{\left(k\right)}\left(x\right)\overline{H{V}_n^{\left(k\right)}\left(x\right)}=2V_n^{\left(k\right)}\left(x\right)\left(V_n^{\left(k\right)}\left(x\right)+H{V}_n^{\left(k\right)}\left(x\right)\right).$$

Theorem 3.2. The recurrence relation of the generalized $k$-order F&L hybrinomials is defined as follows:

$$H{V}_n^{\left(k\right)}\left(x\right)=\sum _{j=1}^k{x}^{k-j}{d}_{j}H{V}_{n-j}^{\left(k\right)}\left(x\right)\mathrm{for}k\ge 5$$

(4)

with the initial conditions

$$\begin{gathered} H{V}_0^{\left(k\right)}\left(x\right)=H{V}_1^{\left(k\right)}\left(x\right)=\dots =H{V}_{k-5}^{\left(k\right)}\left(x\right)=0, \\ H{V}_{k-4}^{\left(k\right)}\left(x\right)=hq,H{V}_{k-3}^{\left(k\right)}\left(x\right)=\epsilon q+h{d}_1{x}^{k-1}, \\ H{V}_{k-2}^{\left(k\right)}\left(x\right)=iq+\epsilon d_1{x}^{k-1}+h\left({d_1}^2{x}^{2(k-1)}+d_{2}q{x}^{k-2}\right), \\ H{V}_{k-1}^{\left(k\right)}\left(x\right)=q+i{d}_1{x}^{k-1}+\epsilon \left({d_1}^2{x}^{2(k-1)}+d_{2}q{x}^{k-2}\right), \\ +h\left(d_1^3{x}^{3(k-1)}+d_1{d}_2{x}^{2k-3}(q+1)+d_{3}q{x}^{k-3}\right), \\ H{V}_k^{\left(k\right)}\left(x\right)=d_1{x}^{k-1}+i\left(d_1^2{x}^{2(k-1)}+d_{2}q{x}^{k-2}\right)+\epsilon \left(d_1^3{x}^{3(k-1)}+d_1{d}_2{x}^{2k-3}(q+1)+d_{3}q{x}^{k-3}\right) \\ +h(d_1^4{x}^{4(k-1)}+d_1{d}_2{x}^{3k-4}(q+1+d_1)+d_{3}q{x}^{k-3}+d_{2}q{x}^{2(k-2)}+d_1{d}_3{x}^{2k-4} \\ +d_{4}q{x}^{k-4}). \end{gathered}$$

Proof. It can be proved by using (2) and (3).

In the following theorem, we give some relations between $H{V}_n^{\left(k\right)}\left(x\right)$ and $\overline{H{V}_n^{\left(k\right)}\left(x\right)}$ for every $x\in \mathbb{R}$.

Theorem 3.3. The generating function for the generalized $k$-order F&L hybrinomials $H{V}_n^{\left(k\right)}\left(x\right)$ is

$$\begin{gathered} g^{\left(k\right)}(x,t)=\sum _{n=0}^{\mathrm{\infty }}H{V}_n^{\left(k\right)}\left(x\right)t^n \\ ={\displaystyle \frac{H{V}_0^{\left(k\right)}\left(x\right)+t\left(H{V}_1^{\left(k\right)}\left(x\right)-d_{1}x{}_{k-1}H{V}_0^{\left(k\right)}\left(x\right)\right)}{1-\sum _{j=1}^k{x}^{k-j}{d}_j{t}^j}} \end{gathered}$$

(5)

Proof. Let $g^{\left(k\right)}(x,t)$ be the generating function for the generalized $k$-order F&L hybrinomials. By making some algebraic operations, we get the following formula:

$$\begin{gathered} g^{\left(k\right)}\left(x,t\right)-x^{k-1}{d}_{1}t{g}^{\left(k\right)}\left(x,t\right)-x^{k-2}{d}_2{t}^2{g}^{\left(k\right)}\left(x,t\right)-\cdots -d_k{t}^k{g}^{\left(k\right)}\left(x,t\right) \\ =H{V}_0^{\left(k\right)}\left(x\right)+t\left(H{V}_1^{\left(k\right)}\left(x\right)-x^{k-1}{d}_{1}H{V}_0^{\left(k\right)}\left(x\right)\right) \\ +t^2\left(\begin{array}{c}H{V}_2^{\left(k\right)}\left(x\right)-x^{k-1}{d}_{1}H{V}_1^{\left(k\right)}\left(x\right)\\ -x^{k-2}{d}_{2}H{V}_0^{\left(k\right)}\left(x\right)\end{array}\right) \\ +t^3\left(\begin{array}{c}H{V}_3^{\left(k\right)}\left(x\right)-x^{k-1}{d}_{1}H{V}_2^{\left(k\right)}\left(x\right)\\ -x^{k-2}{d}_{2}H{V}_1^{\left(k\right)}\left(x\right)-x^{k-2}{d}_{3}H{V}_0^{\left(k\right)}\left(x\right)\end{array}\right) \\ +\sum _{n=4}^{\mathrm{\infty }}{t}^n\left(H{V}_n^{\left(k\right)}\left(x\right)-\sum _{j=1}^{n-1}{x}^{k-j}{d}_{j}H{V}_{n-j}^{\left(k\right)}\left(x\right)\right). \end{gathered}$$

Then, we make the necessary arrangements. Thus, we obtain

$$g^{\left(k\right)}(x,t)={\displaystyle \frac{H{V}_0^{\left(k\right)}\left(x\right)+t\left(H{V}_1^{\left(k\right)}\left(x\right)-d_1{x}^{k-1}H{V}_0^{\left(k\right)}\left(x\right)\right)}{1-\sum _{j=1}^k{x}^{k-j}{d}_j{t}^j}}$$

Corollary 3.1. For $x=1$, we obtain the generating function of the generalized $k$-order F&L hybrid numbers in [4] as follows:

$$g^{\left(k\right)}\left(1,t\right)={\displaystyle \frac{H{V}_0^{\left(k\right)}+t\left(H{V}_1^{\left(k\right)}-d_{1}H{V}_0^{\left(k\right)}\right)}{1-\sum _{j=1}^k{d}_j{t}^j}}$$

where $H{V}_n^{(k)}$ is the $nth$ generalized $k-$order F&L hybrid number.

Corollary 3.2. For $k=2$; we get the generating function of Horadam hybrinomials in [19] as follows:

$$g\left(x,t\right)={\displaystyle \frac{H_0\left(x\right)+t\left(H_1\left(x\right)-d_{1}x{H}_0\left(x\right)\right)}{1-d_{1}xt-d_2{t}^2}}.$$

Corollary 3.3. For the case of $k=2$, we get the generating functions of the following hybrinomials depending on the choice of $d_1,d_2$ and $q$ as follows:

• For $d_1=d_2=1$ and $q=1$, Fibonacci hybrinomials in [9],
• For $d_1=d_2=1$ and $q=2$, Lucas hybrinomials in [9],
• For $d_1=2,d_2=1$ and $q=1$, Pell hybrinomails in [13],
• For $d_1=2,d_2=1$ and $q=2$, Pell-Lucas hybrinomails,
• For $d_1=1,d_2=2$ and $q=1$, Jacobsthal hybrinomails in [20],
• For $d_1=1,d_2=2$ and $q=2$, Jacobsthal-Lucas hybrinomails.

Corollary 3.4. For $k=2$ and $x=1$, we obtain the generating functions of the Horadam, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas hybrid numbers.

4. Matrix Representations of the Generalized k-Order Fibonacci and Lucas Hybrinomials

In this section, we define matrix representations of generalized $k$-order F&L hybrinomials. First, we derive $k\times k$ matrices $Q_k\left(x\right),H_k\left(x\right)$ and $E_{k,n}\left(x\right)$ that play similar role to the $Q$-matrix of Fibonacci numbers.

We determine $k\times k$ matrices $Q_k\left(x\right),H_k\left(x\right)$ and $E_{k,n}\left(x\right)$ as follows:

$$Q_k\left(x\right)={\left[\begin{array}{cccccc}{d}_1{x}^{k-1}& d_2{x}^{k-2}& d_3{x}^{k-3}& \cdots & d_{k-1}x& d_k\\ 1& 0& 0& \cdots & 0& 0\\ 0& 1& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0\\ 0& 0& 0& \cdots & 1& 0\end{array}\right]}_{k\times k},$$

$$H_k\left(x\right)={\left[\begin{array}{cccccc}H{V}_{k-1}^{\left(k\right)}\left(x\right)& H{V}_{k-2}^{\left(k\right)}\left(x\right)& H{V}_{k-3}^{\left(k\right)}\left(x\right)& \cdots & H{V}_1^{\left(k\right)}\left(x\right)& H{V}_0^{\left(k\right)}\left(x\right)\\ H{V}_{k-2}^{\left(k\right)}\left(x\right)& H{V}_{k-3}^{\left(k\right)}\left(x\right)& H{V}_{k-4}^{\left(k\right)}\left(x\right)& \cdots & H{V}_0^{\left(k\right)}\left(x\right)& H{V}_{-1}^{\left(k\right)}\left(x\right)\\ H{V}_{k-3}^{\left(k\right)}\left(x\right)& H{V}_{k-4}^{\left(k\right)}\left(x\right)& H{V}_{k-5}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{-1}^{\left(k\right)}\left(x\right)& H{V}_{-2}^{\left(k\right)}\left(x\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ H{V}_1^{\left(k\right)}\left(x\right)& H{V}_0^{\left(k\right)}\left(x\right)& H{V}_{-1}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{3-k}^{\left(k\right)}\left(x\right)& H{V}_{2-k}^{\left(k\right)}\left(x\right)\\ H{V}_0^{\left(k\right)}\left(x\right)& H{V}_{-1}^{\left(k\right)}\left(x\right)& H{V}_{-2}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{2-k}^{\left(k\right)}\left(x\right)& H{V}_{1-k}^{\left(k\right)}\left(x\right)\end{array}\right]}_{k\times k}$$

and

$$E_{k,n}\left(x\right)={\left[\begin{array}{cccccc}H{V}_{n+k-1}^{\left(k\right)}\left(x\right)& H{V}_{n+k-2}^{\left(k\right)}\left(x\right)& H{V}_{n+k-3}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+1}^{\left(k\right)}\left(x\right)& H{V}_n^{\left(k\right)}\left(x\right)\\ H{V}_{n+k-2}^{\left(k\right)}\left(x\right)& H{V}_{n+k-3}^{\left(k\right)}\left(x\right)& H{V}_{n+k-4}^{\left(k\right)}\left(x\right)& \cdots & H{V}_n^{\left(k\right)}\left(x\right)& H{V}_{n-1}^{\left(k\right)}\left(x\right)\\ H{V}_{n+k-3}^{\left(k\right)}\left(x\right)& H{V}_{n+k-4}^{\left(k\right)}\left(x\right)& H{V}_{n+k-5}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n-1}^{\left(k\right)}\left(x\right)& H{V}_{n-2}^{\left(k\right)}\left(x\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ H{V}_{n+1}^{\left(k\right)}\left(x\right)& H{V}_n^{\left(k\right)}\left(x\right)& H{V}_{n-1}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+3-k}^{\left(k\right)}\left(x\right)& H{V}_{n+2-k}^{\left(k\right)}\left(x\right)\\ H{V}_n^{\left(k\right)}\left(x\right)& H{V}_{n-1}^{\left(k\right)}\left(x\right)& H{V}_{n-2}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+2-k}^{\left(k\right)}\left(x\right)& H{V}_{n+1-k}^{\left(k\right)}\left(x\right)\end{array}\right]}_{k\times k}.$$

Lemma 4.1. Let $Q_k\left(x\right)$ and $E_{k,n}\left(x\right)$ be matrices that is defined with the above equalities. Then, we get

$$E_{k,n+1}\left(x\right)=Q_k\left(x\right)E_{k,n}\left(x\right)$$

for $n\ge 1$.

Proof. By multiplying $k\times k$ matrices $Q_k\left(x\right)$ and $E_{k,n}\left(x\right)$, we get a $k\times k$ matrix as following:

$$Q_k\left(x\right)E_{k,n}\left(x\right)={\left[\begin{array}{ccccc}{a}_{\mathrm{1,1}}& a_{\mathrm{1,2}}& \cdots & a_{1,k-1}& a_{1,k}\\ a_{\mathrm{2,1}}& a_{\mathrm{2,2}}& \cdots & a_{2,k-1}& a_{2,k}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ a_{k-\mathrm{1,1}}& a_{k-\mathrm{2,2}}& \cdots & a_{k-1,k-1}& a_{k-1,k}\\ a_{k,1}& a_{k,2}& \cdots & a_{k,k-1}& a_{k,k}\end{array}\right]}_{k\times k}$$

According to the matrix multiplication,

All the other elements of the other rows are found in the same way.

Since

$${\left[\begin{array}{cccccc}H{V}_{n+k}^{\left(k\right)}\left(x\right)& H{V}_{n+k-1}^{\left(k\right)}\left(x\right)& H{V}_{n+k-2}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+2}^{\left(k\right)}\left(x\right)& H{V}_{n+1}^{\left(k\right)}\left(x\right)\\ H{V}_{n+k-1}^{\left(k\right)}\left(x\right)& H{V}_{n+k-2}^{\left(k\right)}\left(x\right)& H{V}_{n+k-3}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+1}^{\left(k\right)}\left(x\right)& H{V}_n^{\left(k\right)}\left(x\right)\\ H{V}_{n+k-2}^{\left(k\right)}\left(x\right)& H{V}_{n+k-3}^{\left(k\right)}\left(x\right)& H{V}_{n+k-4}^{\left(k\right)}\left(x\right)& \cdots & H{V}_n^{\left(k\right)}\left(x\right)& H{V}_{n-1}^{\left(k\right)}\left(x\right)\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ H{V}_{n+2}^{\left(k\right)}\left(x\right)& H{V}_{n+1}^{\left(k\right)}\left(x\right)& H{V}_n^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+4-k}^{\left(k\right)}\left(x\right)& H{V}_{n+3-k}^{\left(k\right)}\left(x\right)\\ H{V}_{n+1}^{\left(k\right)}\left(x\right)& H{V}_n^{\left(k\right)}\left(x\right)& H{V}_{n-1}^{\left(k\right)}\left(x\right)& \cdots & H{V}_{n+3-k}^{\left(k\right)}\left(x\right)& H{V}_{n+2-k}^{\left(k\right)}\left(x\right)\end{array}\right]}_{k\times k}=E_{k,n+1}\left(x\right)$$

we get $Q_k\left(x\right)E_{k,n}\left(x\right)=E_{k,n+1}\left(x\right)$.

Theorem 4.1. Let $Q_k\left(x\right),H_k\left(x\right)$ and $E_{k,n}\left(x\right)$ be matrices that is defined as above. Then,

$$E_{k,n}\left(x\right)=Q_k^n\left(x\right)H_k\left(x\right)$$

for $n\ge 1$.

Proof. This theorem is proved by induction on $n$. It is easily seen that the assertion is true for $n=1$ since

$$E_{k,1}\left(x\right)=Q_k\left(x\right)H_k\left(x\right).$$

Assuming the assertion is true for $n>1$, we have

$$E_{k,n}\left(x\right)=Q_k^n\left(x\right)H_k\left(x\right).$$

By multiplying each side of this equality with $Q_k\left(x\right)$, we get

$$Q_k\left(x\right)E_{k,n}\left(x\right)=Q_k^{n+1}\left(x\right)H_k\left(x\right)$$

Using Lemma 4.1, we obtain

$$E_{k,n+1}\left(x\right)=Q_k^{n+1}\left(x\right).H_k\left(x\right).$$

Thus, the proof is completed.

Corollary 4.1. For $x=1$, we get the matrix representation of generalized $k$-order F&L hybrid numbers that is shown in [4].

Corollary 4.2. For $k=2$, we can show the matrix representation of Horadam hybrinomials in [12] as

$$Q_2^n\left(x\right)H_2\left(x\right)={\left[\begin{array}{cc}{d}_{1}x& d_2\\ 1& 0\end{array}\right]}^n\left[\begin{array}{cc}H{V}_1^{\left(2\right)}\left(x\right)& H{V}_0^{\left(2\right)}\left(x\right)\\ H{V}_0^{\left(2\right)}\left(x\right)& H{V}_{-1}^{\left(2\right)}\left(x\right)\end{array}\right]=\left[\begin{array}{cc}H{V}_{n+1}^{\left(2\right)}\left(x\right)& H{V}_n^{\left(2\right)}\left(x\right)\\ H{V}_n^{\left(2\right)}\left(x\right)& H{V}_{n-1}^{\left(2\right)}\left(x\right)\end{array}\right]=E_{2,n}\left(x\right).$$

Corollary 4.3. For $k=2$, we can get some special matrix representations as follows:

• For $d_1=d_2=1$ and $q=1$, Fibonacci hybrinomials,
• For $d_1=d_2=1$ and $q=2$, Lucas hybrinomials,
• For $d_1=2,d_2=1$ and $q=1$, Pell hybrinomials,
• For $d_1=2,d_2=1$ and $q=2$, Pell-Lucas hybrinomials.
• For $d_1=1,d_2=2$ and $q=1$, Jacobsthal hybrinomials,
• For $d_1=1,d_2=2$ and $q=2$, Jacobsthal-Lucas hybrinomials.

Corollary 4.4. For $k=3$, $d_1=d_2=d_3=1$ and $q=1$, we can show the matrix representation of Tribonacci hybrinomials as follows:

$$Q_3^n\left(x\right)H_3\left(x\right)={\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]}^n.\left[\begin{array}{ccc}H{V}_2^{\left(3\right)}\left(x\right)& H{V}_1^{\left(3\right)}\left(x\right)& H{V}_0^{\left(3\right)}\left(x\right)\\ H{V}_1^{\left(3\right)}\left(x\right)& H{V}_0^{\left(3\right)}\left(x\right)& H{V}_{-1}^{\left(3\right)}\left(x\right)\\ H{V}_0^{\left(3\right)}\left(x\right)& H{V}_{-1}^{\left(3\right)}\left(x\right)& H{V}_{-2}^{\left(3\right)}\left(x\right)\end{array}\right] =\left[\begin{array}{ccc}H{V}_{n+2}^{\left(3\right)}\left(x\right)& H{V}_{n+1}^{\left(3\right)}\left(x\right)& H{V}_n^{\left(3\right)}\left(x\right)\\ H{V}_{n+1}^{\left(3\right)}\left(x\right)& H{V}_n^{\left(3\right)}\left(x\right)& H{V}_{n-1}^{\left(3\right)}\left(x\right)\\ H{V}_n^{\left(3\right)}\left(x\right)& H{V}_{n-1}^{\left(3\right)}\left(x\right)& H{V}_{n-2}^{\left(3\right)}\left(x\right)\end{array}\right].$$

Also, when we take $x=1$ in this notation, we get the matrix representation of Tribonacci hybrid numbers.

Corollary 4.5. For $k=2$ and $x=1$, we get the matrix representations of the Horadam, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas hybrid numbers.

Corollary 4.6. Let for $n\ge 1$. Then we get as

$${\left[\begin{array}{cccccc}{d}_1{x}^{k-1}& d_2{x}^{k-2}& d_3{x}^{k-3}& \cdots & d_{k-1}x& d_k\\ 1& 0& 0& \cdots & 0& 0\\ 0& 1& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 0& 0\\ 0& 0& 0& \cdots & 1& 0\end{array}\right]}^n\left[\begin{array}{c}H{V}_{k-1}^{\left(k\right)}\left(x\right)\\ H{V}_{k-2}^{\left(k\right)}\left(x\right)\\ H{V}_{k-3}^{\left(k\right)}\left(x\right)\\ ⋮\\ H{V}_1^{\left(k\right)}\left(x\right)\\ H{V}_0^{\left(k\right)}\left(x\right)\end{array}\right]=\left[\begin{array}{c}H{V}_{n+k-1}^{\left(k\right)}\left(x\right)\\ H{V}_{n+k-2}^{\left(k\right)}\left(x\right)\\ H{V}_{n+k-3}^{\left(k\right)}\left(x\right)\\ ⋮\\ H{V}_{n+1}^{\left(k\right)}\left(x\right)\\ H{V}_n^{\left(k\right)}\left(x\right)\end{array}\right].$$

5. Conclusions

This study consists of four parts. In the first part of this study, Fibonacci numbers, Fibonacci polynomials and previous studies about the topic that we consider in this article were mentioned. In addition, the definition of hybrid numbers was made and the multiplication table was given.

In the second part of this study, we defined the generalized $k$-order F&L polynomials by using the generalized $k$-order F&L numbers that we have defined earlier. For the special cases of $d_1,d_2,d_3,...,d_k,q$ and $k$, we show that one can obtain Horadam, Fibonacci, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers and their polynomials. By changing these selections, we can obtain other special polynomials and numbers.

In the third part of this study, we defined the generalized $k$-order F&L hybrinomials using the generalized $k$-order F&L polynomials. Besides, we described the recurrence relations of the generalized $k$-order F&L hybrinomials. With special choices in this recurrence relation, one can obtain the hybrid polynomials such as Horadam, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas hybrinomials. Also, we introduced the generating functions of hybrinomials and gave some important properties of hybrinomials.

The last part of this study, we gave matrix representations of the generalized $k$-order F&L hybrinomials. We show that by special choices of the integers $d_i$ and $q$, one can obtain matrix representations of some known hybrid polynomials such as Horadam, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas hybrinomials.

As a result, by changing the integers $d_i$ and $q$ that we is used in these definitions, we can define many known special polynomials and hybrinomials such as Balancing, Chebyshev hybrinomials etc.