A Note on Generalized k-Order F&L Hybrinomials

Abstract

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the k-order. Hybrid polynomials defined by this generalization are called k-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized k-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized k-order F&L hybrinomials and give some properties of the matrix representations.

1. Introduction

Special numbers and polynomials are frequently discussed in studies in the fields of number theory, applied mathematics, combinatorial probability, mathematical analysis and mathematical physics. Recently, several researchers have examined the properties of these combinatorial numbers and polynomials, presenting a range of relations involving those numbers and polynomials using their generating functions (see, for instance, [1,2,3,4,5]). In addition, generating functions for the families of special polynomials have been developed in [6,7].

One of the most well-known families of special numbers consists of F&L numbers. Fibonacci numbers are defined by the recurrence relation of $F_n=F_{n-1}+F_{n-2}$, with the initial conditions $F_0=0 , F_1=1$ for $n\ge 2$. The recurrence relation of Lucas numbers is the same as that of Fibonacci numbers but differs in terms of initial conditions (see, for details, [8,9,10]). The matrix representation of the Fibonacci sequence, known as the Fibonacci $Q$-matrix (see, for details, [11]), is one of the most significant properties of this sequence and is given 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, the $n\mathrm{t}\mathrm{h}$-power of the Fibonacci $Q$-matrix is shown in [12] by

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

Just like special numbers, the family of Fibonacci polynomials is the most well-known family of special functions. Fibonacci polynomials were initially examined 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 defined by Catalan is the one that is known as the Fibonacci polynomial. Later, all these different definitions were named F&L-type polynomials.

Apart from special numbers and polynomials, the idea of hybrid numbers and polynomials is another concept that has gained increasing prominence not just in mathematics but also in physics and engineering. Ulrych explored the potential of hyperbolic numbers in relativistic quantum physics, demonstrating their use as a generalization of complex numbers within this framework in [13]. This work highlights the potential for hybrid numbers to offer new perspectives and tools for understanding quantum phenomena. Also, Branicky 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 in [14].

Hybrid numbers are briefly defined as generalizations of complex, hyperbolic and dual numbers. The non-commutative number system including these three number systems was defined in [15] as follows:

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

(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 operations of equality, addition, subtraction and multiplication by a scalar are defined as follows:

Equality:	$P=Q$ only if $a_1=a_2,b_1=b_2,c_1=c_2,d_1=d_2.$
Addition:	$P+Q=\left(a_1+a_2\right)+\left(b_1+b_2\right)i+\left(c_1+c_2\right)\epsilon +\left(d_1+d_2\right)h.$
Subtraction:	$P-Q=\left(a_1-a_2\right)+\left(b_1-b_2\right)i+\left(c_1-c_2\right)\epsilon +\left(d_1-d_2\right)h.$
Multiplication by scalar $k\in \mathbb{R}:$	$kP=k{a}_1+k{b}_{1}i+k{c}_1\mathcal{E}+k{d}_{1}h.$

The multiplication of hybrid numbers is defined in Table 1 using Equation (1) as follows:

Table 1. The multiplication of hybrid numbers.

$.$	$1$	$\mathit{i}$	$\mathit{\epsilon }$	$\mathit{h}$
$i$	$i$	$-1$	$1-h$	$\epsilon +i$
$\epsilon$	$\epsilon$	$h+1$	$0$	$-\epsilon$
$h$	$h$	$-\epsilon -i$	$\epsilon$	$1$

Horadam hybrid numbers were introduced in [16] for the first time. Later, in [17], Fibonacci hybrid numbers were studied and some important properties about the hybrid numbers were presented (see, for details, [18,19,20]). Besides these, Pell and Pell–Lucas hybrid numbers and Jacobsthal and Jacobsthal–Lucas hybrid numbers were introduced in [21,22] by using the terms of these sequences. Further details on these sequences can be found in [23,24,25,26]. Later, third-order Jacobsthal and Jacobsthal hybrinomials were introduced in [27]. A new generalization based on these studies was defined in [28]. Also, some algebraic properties regarding F&L hybrid numbers were given in this reference. In [29], these studies were generalized, and generalized k-order F&L hybrid numbers were defined.

Motivated by the above studies, we introduce in this paper a generalization of k-order F&L polynomials, called generalized k-order F&L hybrinomials. The significance of these hybrinomials is that they provide a natural generalization of numerous well-known hybrinomials, including Jacobsthall and Jacobsthall–Lucas hybrinomials, Horadam hybrinomials, Fibonacci and Lucas hybrinomials, Pell and Pell–Lucas hybrinomials and others. Thus, one of the most thorough generalizations of this topic is provided. Furthermore, various properties of generalized k-order F&L hybrinomials including recurrence relations and generating functions are derived, and their matrix representations are presented.

2. Generalized k-Order F&L Polynomials

In this section, we first introduce generalized k-order F&L polynomials that are defined using generalized k-order F&L numbers. Then, we give some special cases of generalized k-order F&L polynomials such as Fibonacci and Lucas polynomials, Pell and Pell–Lucas polynomials and Jacobsthal and Jacobsthal–Lucas polynomials.

Before presenting our results, we recall the definition of generalized k-order F&L numbers.

Generalized k-order F&L numbers are defined by the following recurrence relation:

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

with initial conditions $V_1^{\left(k\right)}=V_2^{\left(k\right)}=V_3^{\left(k\right)}=\dots =V_{k-2}^{\left(k\right)}=0,V_{k-1}^{\left(k\right)}=q,V_k^{\left(k\right)}=d_1$ where $d_i(1\le i\le k)$ and $q$ are integers [29].

Definition 1.

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

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

(2)

with initial conditions

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

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

Table 2. Special polynomials.

${\mathit{d}}_1$	${\mathit{d}}_2$	$\mathit{q}$	Special Polynomials
1	1	2	Lucas polynomials
2	1	1	Pell polynomials
2	1	2	Pell–Lucas polynomials
1	2	1	Jacobsthal polynomials
1	2	2	Jacobsthal–Lucas polynomials

• For $k=2$, we obtain the following table:
• For $k=3$ and $d_1=d_2=d_3=1,q=1$, we obtain 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 $T_1\left(x\right)=0,T_2\left(x\right)=1,T_3\left(x\right)=x^2$.
• For $d_1=d_2=d_3=.\cdots =d_k=1$ and $q=1$, we obtain $k$-order Fibonacci polynomials.
• For $d_1=d_2=d_3=\cdots =d_k=1$ and $q=2$, we obtain $k$-order Lucas polynomials.
• For $d_1=2,d_2=d_3=\cdots =d_k=1$ and $q=1$, we obtain $k$-order Pell polynomials.
• For $d_1=2,d_2=d_3=\cdots =d_k=1$ and $q=2$, we obtain $k$-order Pell–Lucas polynomials.
• For $d_1=1,d_2=2,d_3\cdots =d_k=1$ and $q=1$, we obtain $k$-order Jacobsthal polynomials.
• For $d_1=1,d_2=2,d_3=\cdots =d_k=1$ and $q=2$, we obtain $k$-order Jacobsthal–Lucas polynomials.

3. Generalized k-Order Fibonacci and Lucas Hybrinomials

In this section, we define generalized $k$-order F&L hybrinomials using generalized $k$-order F&L polynomials. We provide recurrence relations, generating functions and some other properties of these hybrinomials.

Definition 2.

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

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

(3)

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

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

• For $k=2$, we obtain some special hybrinomials which are included in [20] using (2) as in Table 3.
  Table 3. Special hybrinomials.

  ${\mathit{d}}_1$	${\mathit{d}}_2$	$\mathit{q}$	Special Hybrinomials
  1	1	1	Fibonacci hybrinomials
  1	1	2	Lucas hybrinomials
  2	1	1	Pell hybrinomials
  2	1	2	Pell–Lucas hybrinomials
  1	2	1	Jacobsthal hybrinomials
  1	2	2	Jacobsthal–Lucas hybrinomials

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

Definition 3.

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

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

Theorem 1.

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

i.

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

ii.

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

iii.

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

  where  $k\ge 2$  and  $n>0.$

Proof. 

i. The proof is easily seen using the definitions of $\mathrm{H}{\mathrm{V}}_{\mathrm{n}}^{(\mathrm{k})}(\mathrm{x})$ and $\overline{\mathrm{H}{\mathrm{V}}_{\mathrm{n}}^{(\mathrm{k})}(\mathrm{x})}$.

ii.

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

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

iii.

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

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

Then, using (ii), we obtain

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

Thus, we obtain

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

□

Theorem 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

$$H{V}_0^{\left(k\right)}\left(x\right)=H{V}_1^{\left(k\right)}\left(x\right)=\dots =H{V}_{k-5}^{\left(k\right)}\left(x\right)=0,$$

$$H{V}_{k-4}^{(k)}(x)=hq,H{V}_{k-3}^{(k)}(x)=\epsilon q+h{d}_1{x}^{k-1},$$

$$H{V}_{k-2}^{(k)}(x)=iq+\epsilon d_1{x}^{k-1}+h\left({d_1}^2{x}^{2(k-1)}+d_{2}q{x}^{k-2}\right),$$

$$\begin{array}{ll}H{V}_{k-1}^{(k)}(x)=q+& i{d}_1{x}^{k-1}+\epsilon \left({d_1}^2{x}^{2(k-1)}+d_{2}q{x}^{k-2}\right)\\ & +h\left(d_1^3{x}^{3(k-1)}+d_1{d}_2{x}^{2k-3}(q+1)+d_{3}q{x}^{k-3}\right),\end{array}$$

$$\begin{array}{ll}H{V}_k^{(k)}(x)=d_1{x}^{k-1}& +i\left(d_1^2{x}^{2(k-1)}+d_{2}q{x}^{k-2}\right)\\ & +\epsilon \left(d_1^3{x}^{3(k-1)}+d_1{d}_2{x}^{2k-3}(q+1)+d_{3}q{x}^{k-3}\right)+h(d_1^4{x}^{4(k-1)}\\ & +d_1{d}_2{x}^{3k-4}(q+1+d_1)+d_{3}q{x}^{k-3}+d_{2}q{x}^{2(k-2)}\\ & +d_1{d}_3{x}^{2k-4}+d_{4}q{x}^{k-4}).\end{array}$$

Proof. 

This can be easily proved by using (2) and (3). □

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

Theorem 3.

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

$$g^{(k)}(x,t)=\sum _{n=0}^{\infty }H{V}_n^{(k)}(x)t^n={\displaystyle \frac{H{V}_0^{(k)}(x)+t\left(H{V}_1^{(k)}(x)-d_1{x}^{k-1}H{V}_0^{(k)}(x)\right)}{1-{\sum }_{j=1}^k{x}^{k-j}{d}_j{t}^j}}$$

(5)

Proof. 

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

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

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

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

□

Corollary 1.

For $x=1$, we obtain the generating function of the generalized  $k$-order F&L hybrid numbers in [29] 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 2.

For  $k=2$, we obtain the generating function of Horadam hybrinomials in [18] 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.

For the case of $k=2$, we obtain 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 [17];
• For $d_1=d_2=1$ and $q=2$, Lucas hybrinomials in [17];
• For $d_1=2,d_2=1$ and $q=1$, Pell hybrinomials in [21];
• For $d_1=2,d_2=1$ and $q=2$, Pell–Lucas hybrinomials in [21];
• For $d_1=1,d_2=2$ and $q=1$, Jacobsthal hybrinomials in [22];
• For $d_1=1,d_2=2$ and $q=2$, Jacobsthal–Lucas hybrinomials in [22].

Corollary 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 $Q_k(x),H_k(x)$ and $E_{k,n}(x)$ that play similar roles to the $Q$-matrix of Fibonacci numbers.

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

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

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

and

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

Lemma 1.

Let  $Q_k(x)$ and  $E_{k,n}(x)$ be matrices that are defined with the above equalities. Then, we obtain

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

for $n\ge 1$.

Proof. 

By multiplying $k\times k$ matrices $Q_k(x)$ and $E_{k,n}(x)$, we obtain a $k\times k$ matrix as follows:

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

According to the matrix multiplication,

❖

For the first row, we obtain the following:

$$\begin{gathered} a_{\mathrm{1,1}}=d_1{x}^{k-1}H{V}_{n+k-1}^{\left(k\right)}\left(x\right)+d_2{x}^{k-2}H{V}_{n+k-2}^{\left(k\right)}\left(x\right)+\cdots +d_{k-1}xH{V}_{n+1}^{(k)}(x) \\ +d_{k}H{V}_n^{(k)}(x)=H{V}_{n+k}^{(k)}(x) \end{gathered}$$

$$\begin{gathered} a_{\mathrm{1,2}}=d_1{x}^{k-1}H{V}_{n+k-2}^{(k)}(x)+d_2{x}^{k-2}H{V}_{n+k-3}^{(k)}(x)+\cdots +d_{k-1}xH{V}_n^{(k)}(x) \\ +d_{k}H{V}_{n-1}^{(k)}(x)=H{V}_{n+k-1}^{(k)}(x) \end{gathered}$$

$$⋮$$

$$\begin{array}{ll}{a}_{1,k}=d_1{x}^{k-1}H{V}_n^{(k)}& (x)\\ & +d_2{x}^{k-2}H{V}_{n-1}^{(k)}(x)+...+d_{k-1}xH{V}_{n+2-k}^{(k)}(x){+d}_{k}H{V}_{n+1-k}^{(k)}(x)\\ & =H{V}_{n+1}^{(k)}(x)\end{array}$$

❖

For the second row, we obtain the following:

$$a_{\mathrm{2,1}}=H{V}_{n+k-1}^{(k)}(x)$$

$$a_{\mathrm{2,2}}=H{V}_{n+k-2}^{(k)}(x)$$

$$⋮$$

$$a_{2,k}=H{V}_n^{(k)}(x)$$

❖

For the $k$th row, we obtain the following:

$$a_{k,1}=H{V}_{n+1}^{(k)}(x)$$

$$a_{k,2}=H{V}_n^{(k)}(x)$$

$$⋮$$

$$a_{k,k}=H{V}_{n-k+2}^{(k)}(x).$$

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

Since

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

we obtain $Q_k(x)E_{k,n}(x)=E_{k,n+1}(x)$. □

Theorem 4.

Let $Q_k(x),H_k(x)$ and $E_{k,n}(x)$ be matrices that are defined as above. Then,

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

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

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

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

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

By multiplying each side of this equality with $Q_k(x)$, we obtain

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

Using Lemma 1, we obtain

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

Thus, the proof is completed. □

Corollary 5.

For $x=1$, we obtain the matrix representations of generalized $k$-order F&L hybrid numbers that are shown in [29].

Corollary 6.

For  $k=2$, we can show the matrix representations of Horadam hybrinomials in [18] as

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

Corollary 7.

For $k=2$, we can obtain 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 8.

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

$$\begin{array}{cc}{Q}_3^n\left(x\right)H_3\left(x\right)& ={\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right]}^n.\left[\begin{array}{ccc}H{V}_2^{\left(3\right)}\left(x\right)& H{V}_1^{\left(3\right)}\left(x\right)& H{V}_0^{\left(3\right)}\left(x\right)\\ H{V}_1^{\left(3\right)}\left(x\right)& H{V}_0^{\left(3\right)}\left(x\right)& H{V}_{-1}^{\left(3\right)}\left(x\right)\\ H{V}_0^{\left(3\right)}\left(x\right)& H{V}_{-1}^{\left(3\right)}\left(x\right)& H{V}_{-2}^{\left(3\right)}\left(x\right)\end{array}\right]\\ & =\left[\begin{array}{ccc}H{V}_{n+2}^{(3)}(x)& H{V}_{n+1}^{(3)}(x)& H{V}_n^{(3)}(x)\\ H{V}_{n+1}^{(3)}(x)& H{V}_n^{(3)}(x)& H{V}_{n-1}^{(3)}(x)\\ H{V}_n^{(3)}(x)& H{V}_{n-1}^{(3)}(x)& H{V}_{n-2}^{(3)}(x)\end{array}\right].\end{array}$$

Furthermore, by taking  $x=1$  in this notation, we obtain the matrix representation of Tribonacci hybrid numbers.

Corollary 9.

For $k=2$ and $x=1$, we obtain the matrix representations of the Horadam, Fibonacci, Lucas, Pell, Pell–Lucas, Jacobsthal and Jacobsthal–Lucas hybrid numbers.

Corollary 10.

For $n\ge 1$, we obtain the following:

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

5. Conclusions

This work is intended as an attempt to introduce a generalization of k-order F&L polynomials, called generalized k-order F&L hybrinomials. Since these hybrinomials provide a natural generalization of well-known hybrinomials, one of the most comprehensive generalizations on this subject is presented in this paper. Furthermore, recurrence relations, generating functions, matrix representations and some other properties of these hybrinomials have been given.

For future research, additional properties regarding these hybrinomials can be defined along with their matrix representations. In addition, generating functions for certain families of special polynomials can be developed.