A New Hybrid Generalization of Balancing Polynomials

Abstract

In this paper, we introduce and study balancing hybrinomials, i.e., polynomials being a generalization of balancing hybrid numbers. We provide some properties of the balancing hybrinomials, including Catalan, Cassini, d’Ocagne, and Vajda identities, among others. Moreover, we present a matrix representation of the hybrinomials.

1. Introduction

Balancing numbers were introduced in [1]. Behera and Panda defined balancing number n as the solution of the Diophantine equation:

$$1+2+\dots +(n-1)=(n+1)+(n+2)+\dots +(n+r).$$

(1)

In this case, an integer r is called a balancer of n. The balancing numbers are denoted by $B_n$. In [1], the authors defined balancing numbers recurrently, using the linear homogeneous recursive equation of order 2:

$$B_n=6B_{n-1}-B_{n-2}\mathrm{for}n\ge 2$$

(2)

with initial terms $B_0=0$ and $B_1=1$.

In [1], it was proved that n is a balancing number if and only if $n^2$ is a triangular number, i.e., $8n^2+1$ is a perfect square. In [2], the author introduced Lucas-balancing numbers, defined as follows: if $B_n$ is a balancing number, then the number $C_n=\sqrt{8{B_n}^2+1}$ is a Lucas-balancing number.

The Lucas-balancing numbers can be defined recurrently using the same relation as for balancing numbers but with different initial conditions:

$$C_n=6C_{n-1}-C_{n-2}\mathrm{for}n\ge 2$$

with $C_0=1$ and $C_1=3$. The characteristic equation of the recurrence relation (2) has the form

$${\lambda }^2-6\lambda +1=0$$

with roots

$${\lambda }_1=3+2\sqrt{2},{\lambda }_2=3-2\sqrt{2}.$$

Balancing numbers and Lucas-balancing numbers satisfy the following relations, called Binet type formulas:

$$B_n={\displaystyle \frac{{{\lambda }_1}^n-{{\lambda }_2}^n}{{\lambda }_1-{\lambda }_2}},$$

$$C_n={\displaystyle \frac{{{\lambda }_1}^n+{{\lambda }_2}^n}{2}}.$$

By modifying the Diophantine Equation (1), new numbers of the balancing type were defined as follows: cobalancing numbers and Lucas-cobalancing numbers. Some interesting properties of these numbers are given in [1,3,4,5]. In [3], the authors using Binet type formulas to balancing type numbers received some new identities and properties for these numbers, including Catalan, Cassini, and d’Ocagne identities, among others. Importantly, the main purpose of the paper [3] was to present the formulas of balancing type numbers in such a way that their possible applications would be facilitated. Cobalancing numbers and cobalancers were introduced in [4]. In [5], further properties of balancing and cobalancing numbers were given. Moreover, the author defined higher order balancing and cobalancing numbers. As with any sequence defined by the recurrence relations, we can also define balancing type numbers with negative indices. For example, for balancing numbers, we have $B_{-n}=-B_n$. Hence, we obtain the sequence for balancing numbers of the form $\dots ,-35,-6,-1,0,1,6,35,\dots$. In [6], the author pointed out the fact that the sequence has a property of symmetry.

Balancing numbers have many applications. In [7], Gautam showed an application of balancing numbers in solving the Diophantine equation. The author found the solutions of equation $x^2+{(x+1)}^2=y^2$ in terms of balancing numbers. Some benefits of using balancing numbers to solve generalized Pell’s equation were presented in [8]. The solution sets of some equations obtained by the balancing numbers concept and the standard method were the same, but the concept of using balancing numbers was much easier. The use of recursive matrices in relation to balancing numbers in cryptography was discussed in [9].

Many kinds of generalizations of balancing and Lucas-balancing numbers are considered in the literature; see [10,11,12,13,14,15,16].

In [10], the authors generalized balancing numbers defining, for fixed positive integers k and l, $(k,l)$-balancing numbers. Moreover, several finiteness results (effective and ineffective) were proved. The $(k,l)$-balancing numbers are also called $(k,l)$-power numerical centers. Another interesting generalization of balancing numbers includes $(a,b)$-balancing numbers, as studied in [11]. In this case, a is positive, b is nonnegative, and a and b are coprime integers. Interestingly, in the proofs of the presented results, the authors combined many known methods (the Baker’s method, modular method, and Chabauty method) and the theory of elliptic curves.

One-parameter generalizations of balancing numbers were investigated in [12,13,14]. In [12], Dash, Ota, and Dash defined the t-balancing numbers and obtained their properties. In [13], Tekcan, Tayat, and Özbek considered the integer solutions of some Diophantine equations in order to determine the general terms of all t-balancing numbers. Note that t-balancing numbers generalize balancing and cobalancing numbers simultaneously. For positive number k, Ray in [14] defined k-balancing numbers and then presented the natural extension of these numbers.

In the case when k is a real variable, we obtain the definition of balancing polynomials. Algebraic properties of k-balancing numbers were presented in [15]. Determinants of some tridiagonal matrices were also used in the proofs of some properties of k-balancing numbers in [16]. Aspects of balancing type numbers can also be related to hypercomplex numbers. The calculations related to them also yield interesting results. In [17,18], balancing quaternions and Lucas-balancing quaternions were introduced independently. The authors presented many interesting properties of these numbers. In [6], balancing split quaternions were defined. In [19], the authors considered bi-periodic balancing quaternions.

Moreover, in the literature, many studies have been conducted on complex, dual-complex, dual hyperbolic, and bihyperbolic numbers, with the coefficients being balancing numbers or some generalizations of these numbers; see [20,21,22,23,24]. Several identities involving these numbers were presented in these papers. As we mentioned earlier, balancing polynomials are a natural extension of balancing numbers.

Several properties of interesting Catalan-Daehee polynomials related to other known numbers are presented in [25,26]. In [25], the authors expressed those polynomials in terms of special polynomials, like Bernoulii polynomials and Euler polynomials. In [26], degenerate Catalan-Daehee polynomials of order r are studied.

Balancing polynomials were introduced in [14] in the following way:

$$B_n\left(x\right)=6x{B}_{n-1}\left(x\right)-B_{n-2}\left(x\right)\mathrm{for}n\ge 2$$

with initial conditions $B_0\left(x\right)=0,$ $B_1\left(x\right)=1.$

In [27], Lucas-balancing polynomials were considered; they are defined in the following way:

$$C_n\left(x\right)=6x{C}_{n-1}\left(x\right)-C_{n-2}\left(x\right)\mathrm{for}n\ge 2$$

with initial conditions $C_0\left(x\right)=1$ and $C_1\left(x\right)=3x.$

It is easy to check whether for $x=1$, we obtain $B_n\left(x\right)=B_n$ and $C_n\left(x\right)=C_n$.

Using the above equalities, we obtain the following:

$$\begin{array}{cc}\hfill B_0\left(x\right)& =0\hfill \\ \hfill B_1\left(x\right)& =1\hfill \\ \hfill B_2\left(x\right)& =6x\hfill \\ \hfill B_3\left(x\right)& =36x^2-1\hfill \\ \hfill B_4\left(x\right)& =216x^3-12x\hfill \\ \hfill \dots & \\ \hfill C_0\left(x\right)& =1\hfill \\ \hfill C_1\left(x\right)& =3x\hfill \\ \hfill C_2\left(x\right)& =18x^2-1\hfill \\ \hfill C_3\left(x\right)& =108x^3-9x\hfill \\ \hfill C_4\left(x\right)& =648x^4-72x^2+1\hfill \\ \hfill \dots & \end{array}$$

Binet type formulas for the balancing polynomials and Lucas-balancing polynomials are of the form

$$B_n\left(x\right)={\displaystyle \frac{{\lambda }^n\left(x\right)-{\lambda }^{-n}\left(x\right)}{\lambda \left(x\right)-{\lambda }^{-1}\left(x\right)}},$$

(3)

$$C_n\left(x\right)={\displaystyle \frac{1}{2}}\left({\lambda }^n\left(x\right)+{\lambda }^{-n}\left(x\right)\right),$$

where $\lambda \left(x\right)=3x+\sqrt{9x^2-1}$ and ${\lambda }^{-1}\left(x\right)=3x-\sqrt{9x^2-1}.$

We will use the following results.

Theorem 1

([28]). Let n be an integer where $n\ge 1$. Then,

$$C_n\left(x\right)=B_{n+1}\left(x\right)-3x{B}_n\left(x\right),$$

(4)

$$C_n\left(x\right)={\displaystyle \frac{1}{2}}(B_{n+1}\left(x\right)-B_{n-1}\left(x\right)),$$

(5)

$$C_n\left(x\right)=3x{B}_n\left(x\right)-B_{n-1}\left(x\right),$$

(6)

$$C_n\left(x\right)=3x{C}_{n-1}\left(x\right)+(9x^2-1)B_{n-1}\left(x\right).$$

(7)

In this paper, we use the concept of balancing and Lucas-balancing polynomials in the theory of hybrid numbers.

In [29], Özdemir introduced the set of hybrid numbers as a generalization of complex, dual, and hyperbolic numbers.

A hybrid number $\mathbf{Z}$ has the form $\mathbf{Z}=a+b\mathbf{i}+c\mathsf{\epsilon }+d\mathbf{h}$, where $a,b,c,d\in \mathbb{R}$, and $\mathbf{i},$$\mathsf{\epsilon },$$\mathbf{h}$ are operators which satisfy the following relations:

$${\mathbf{i}}^2=-1,{\mathsf{\epsilon }}^2=0,{\mathbf{h}}^2=1,\mathbf{i}\mathbf{h}=-\mathbf{h}\mathbf{i}=\mathsf{\epsilon }+\mathbf{i}.$$

(8)

The set of hybrid numbers is denoted by $\mathbb{K}$. Let ${\mathbf{Z}}_1=a_1+b_1\mathbf{i}+c_1\mathsf{\epsilon }+d_1\mathbf{h}$ and ${\mathbf{Z}}_2=a_2+b_2\mathbf{i}+c_2\mathsf{\epsilon }+d_2\mathbf{h}$ be any two hybrid numbers. Then,

${\mathbf{Z}}_1={\mathbf{Z}}_2$ if and only if $a_1=a_2,$ $b_1=b_2,$ $c_1=c_2,$ $d_1=d_2,$

${\mathbf{Z}}_1+{\mathbf{Z}}_2=(a_1+a_2)+(b_1+b_2)\mathbf{i}+(c_1+c_2)\mathsf{\epsilon }+(d_1+d_2)\mathbf{h},$

${\mathbf{Z}}_1-{\mathbf{Z}}_2=(a_1-a_2)+(b_1-b_2)\mathbf{i}+(c_1-c_2)\mathsf{\epsilon }+(d_1-d_2)\mathbf{h},$

for $\alpha \in \mathbb{R}$   $\alpha {\mathbf{Z}}_1=\alpha a_1+\alpha b_1\mathbf{i}+\alpha c_1\mathsf{\epsilon }+\alpha d_1\mathbf{h}.$

Using (8), we can multiply hybrid numbers.

Table 1. The hybrid number multiplication.

·	i	$\mathsf{\epsilon }$	h
$\mathbf{i}$	$-1$	$1-\mathbf{h}$	$\mathsf{\epsilon }+\mathbf{i}$
$\mathsf{\epsilon }$	$\mathbf{h}+1$	0	$-\mathsf{\epsilon }$
$\mathbf{h}$	$-\mathsf{\epsilon }-\mathbf{i}$	$\mathsf{\epsilon }$	1

presents products of operators $\mathbf{i}$, $\mathsf{\epsilon },$ and $\mathbf{h}$.

It is easy to see that the multiplication of hybrid numbers can be carried out in the same way as the multiplication of algebraic expressions. Other properties of hybrid numbers are given in [29]. In [30], a special kind of hybrid number (balancing hybrid numbers) was introduced in the following way:

The nth balancing hybrid number $B{H}_n$ and the nth Lucas-balancing hybrid number $C{H}_n$ were defined as

$$B{H}_n=B_n+B_{n+1}\mathbf{i}+B_{n+2}\mathsf{\epsilon }+B_{n+3}\mathbf{h},$$

$$C{H}_n=C_n+C_{n+1}\mathbf{i}+C_{n+2}\mathsf{\epsilon }+C_{n+3}\mathbf{h},$$

respectively.

In [31], some new properties of hybrid balancing numbers were given.

In addition, other types of hybrid numbers, such as cobalancing hybrid numbers, can also be found in the literature; see [32]. The authors considered hybrid numbers with cobalancing and Lucas-cobalancing coefficients, and they received, among others, general formulas which imply the Catalan, Cassini, Vajda, d’Ocagne, and Halton identities.

The term “hybrinomials” was used for the first time in [33]. The authors defined Fibonacci and Lucas hybrinomials as polynomials that are generalizations of Fibonacci hybrid numbers and Lucas hybrid numbers, respectively. Moreover, the authors provided many basic properties of the Fibonacci and Lucas hybrinomials, including Binet’s formulas, some identities, sums of the finite terms of these hybrinomials, etc. It is important to highlight that scientists have embraced the concept of “hybrinomials” and that numerous papers on this topic have been published in recent years.

Further generalizations of hybrinomials can be located in the following references [34,35,36].

In the next section, we will define balancing and Lucas-balancing hybrinomials.

2. Balancing Hybrinomials and Lucas-Balancing Hybrinomials

For an integer $n\ge 0$, balancing hybrinomials and Lucas-balancing hybrinomials are defined by

$$B{H}_n\left(x\right)=B_n\left(x\right)+B_{n+1}\left(x\right)\mathbf{i}+B_{n+2}\left(x\right)\mathsf{\epsilon }+B_{n+3}\left(x\right)\mathbf{h}$$

(9)

and

$$C{H}_n\left(x\right)=C_n\left(x\right)+C_{n+1}\left(x\right)\mathbf{i}+C_{n+2}\left(x\right)\mathsf{\epsilon }+C_{n+3}\left(x\right)\mathbf{h},$$

where $B_n\left(x\right)$ is the nth balancing polynomial, $C_n\left(x\right)$ is the nth Lucas-balancing polynomial, and $\mathbf{i}$, $\mathsf{\epsilon }$, $\mathbf{h}$ are hybrid units which satisfy (8).

If $x=1$, then we obtain the definition of balancing hybrid numbers $B{H}_n$ and Lucas-balancing hybrid numbers $C{H}_n$, respectively.

It is easy to see that balancing hybrinomials are a special case of Horadam hybrinomials, defined in [35]. In this paper, we will present selected properties of balancing hybrinomials and Lucas-balancing hybrinomials. Some properties of balancing hybrinomials are special cases of properties of Horadam hybrinomials, but the properties of Lucas-balancing hybrinomials and the connections between these hybrinomials are new.

Theorem 2.

For any variable quantity x, we have

$$B{H}_n\left(x\right)=6xB{H}_{n-1}\left(x\right)-B{H}_{n-2}\left(x\right)\mathrm{for}n\ge 2$$

(10)

with $B{H}_0\left(x\right)=\mathbf{i}+6x\mathsf{\epsilon }+(36x^2-1)\mathbf{h}$

and $B{H}_1\left(x\right)=1+6x\mathbf{i}+(36x^2-1)\mathsf{\epsilon }+(216x^3-12x)\mathbf{h}.$

Proof. 

Let $n=2$. Then,

$$\begin{array}{cc}\hfill B{H}_2\left(x\right)& =6xB{H}_1\left(x\right)-B{H}_0\left(x\right)\hfill \\ \hfill & =6x(1+6x\mathbf{i}+(36x^2-1)\mathsf{\epsilon }+(216x^3-12x)\mathbf{h})\hfill \\ \hfill & -\mathbf{i}-6x\mathsf{\epsilon }-(36x^2-1)\mathbf{h}\hfill \\ \hfill & =6x+(36x^2-1)\mathbf{i}+(216x^3-12x)\mathsf{\epsilon }+(1296x^4-108x^2+1)\mathbf{h}\hfill \\ \hfill & =B_2\left(x\right)+B_3\left(x\right)\mathbf{i}+B_4\left(x\right)\mathsf{\epsilon }+B_5\left(x\right)\mathbf{h}.\hfill \end{array}$$

Let $n\ge 3$. Using the definition of balancing polynomials, we obtain

$$\begin{array}{cc}\hfill B{H}_n\left(x\right)& =B_n\left(x\right)+B_{n+1}\left(x\right)\mathbf{i}+B_{n+2}\left(x\right)\mathsf{\epsilon }+B_{n+3}\left(x\right)\mathbf{h}\hfill \\ \hfill & =(6x{B}_{n-1}\left(x\right)-B_{n-2}\left(x\right))+(6x{B}_n\left(x\right)-B_{n-1}\left(x\right))\mathbf{i}\hfill \\ \hfill & +(6x{B}_{n+1}\left(x\right)-B_n\left(x\right))\mathsf{\epsilon }+(6x{B}_{n+2}\left(x\right)-B_{n+1}\left(x\right))\mathbf{h}\hfill \\ \hfill & =6x(B_{n-1}\left(x\right)+B_n\left(x\right)\mathbf{i}+B_{n+1}\left(x\right)\mathsf{\epsilon }+B_{n+2}\left(x\right)\mathbf{h})\hfill \\ \hfill & -\left(B_{n-2}\left(x\right)+B_{n-1}\left(x\right)\mathbf{i}+B_n\left(x\right)\mathsf{\epsilon }+B_{n+1}\left(x\right)\mathbf{h}\right)\hfill \\ \hfill & =6xB{H}_{n-1}\left(x\right)-B{H}_{n-2}\left(x\right),\hfill \end{array}$$

which completes the proof. □

In the same way, we can prove the next result.

Theorem 3.

For any variable quantity x, we have

$$C{H}_n\left(x\right)=6xC{H}_{n-1}\left(x\right)-C{H}_{n-2}\left(x\right)\mathrm{for}n\ge 2$$

with $C{H}_0\left(x\right)=1+3x\mathbf{i}+(18x^2-1)\mathsf{\epsilon }+(108x^3-9x)\mathbf{h}$

and $C{H}_1\left(x\right)=3x+(18x^2-1)\mathbf{i}+(108x^3-9x)\mathsf{\epsilon }+(648x^4-72x^2+1)\mathbf{h}.$

Now, we will present some identities for balancing and Lucas-balancing hybrinomials.

Theorem 4.

Let $n\ge 0$ be an integer. Then,

$$C{H}_n\left(x\right)=B{H}_{n+1}\left(x\right)-3xB{H}_n\left(x\right).$$

Proof. 

By Formula (4), we have

$$\begin{array}{cc}\hfill & B{H}_{n+1}\left(x\right)-3xB{H}_n\left(x\right)\hfill \\ \hfill & =B_{n+1}\left(x\right)+B_{n+2}\left(x\right)\mathbf{i}+B_{n+3}\left(x\right)\mathsf{\epsilon }+B_{n+4}\left(x\right)\mathbf{h}\hfill \\ \hfill & -3x\left(B_n\left(x\right)+B_{n+1}\left(x\right)\mathbf{i}+B_{n+2}\left(x\right)\mathsf{\epsilon }+B_{n+3}\left(x\right)\mathbf{h}\right)\hfill \\ \hfill & =B_{n+1}\left(x\right)-3x{B}_n\left(x\right)+(B_{n+2}\left(x\right)-3x{B}_{n+1}\left(x\right))\mathbf{i}\hfill \\ \hfill & +(B_{n+3}\left(x\right)-3x{B}_{n+2}\left(x\right))\mathsf{\epsilon }+(B_{n+4}\left(x\right)-3x{B}_{n+3}\left(x\right))\mathbf{h}\hfill \\ \hfill & =C_n\left(x\right)+C_{n+1}\left(x\right)\mathbf{i}+C_{n+2}\left(x\right)\mathsf{\epsilon }+C_{n+3}\left(x\right)\mathbf{h}=C{H}_n\left(x\right).\hfill \end{array}$$

□

Using (5)–(7), we obtain the following theorem.

Theorem 5.

Let $n\ge 1$ be an integer. Then,

$$C{H}_n\left(x\right)={\displaystyle \frac{1}{2}}(B{H}_{n+1}\left(x\right)-B{H}_{n-1}\left(x\right)),$$

$$C{H}_n\left(x\right)=3xB{H}_n\left(x\right)-B{H}_{n-1}\left(x\right),$$

$$C{H}_n\left(x\right)=3xC{H}_{n-1}\left(x\right)+(9x^2-1)B{H}_{n-1}\left(x\right).$$

3. Binet Type Formulas and Some Properties of Balancing Hybrinomials and Lucas-Balancing Hybrinomials

In this section, we will present Binet type formulas for balancing hybrinomials and Lucas-balancing hybrinomials. Moreover, we will provide Catalan, Cassini, d’Ocagne, and Vajda identities for these hybrinomials.

Theorem 6.

Let $n\ge 0$ be an integer. Then,

$$\begin{array}{cc}\hfill B{H}_n\left(x\right)& {\displaystyle =\frac{{\lambda }^n\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}\left(1+\lambda \left(x\right)\mathbf{i}+{\lambda }^2\left(x\right)\mathsf{\epsilon }+{\lambda }^3\left(x\right)\mathbf{h}\right)}\hfill \\ \hfill & {\displaystyle -\frac{{\gamma }^n\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}\left(1+\gamma \left(x\right)\mathbf{i}+{\gamma }^2\left(x\right)\mathsf{\epsilon }+{\gamma }^3\left(x\right)\mathbf{h}\right),}\hfill \end{array}$$

(11)

where $\lambda \left(x\right)=3x+\sqrt{9x^2-1}$ and $\gamma \left(x\right)={\lambda }^{-1}\left(x\right)=3x-\sqrt{9x^2-1}.$

Proof. 

Using (9) and (3), we have

$$\begin{array}{cc}\hfill B{H}_n\left(x\right)& =B_n\left(x\right)+B_{n+1}\left(x\right)\mathbf{i}+B_{n+2}\left(x\right)\mathsf{\epsilon }+B_{n+3}\left(x\right)\mathbf{h}\hfill \\ \hfill & {\displaystyle =\frac{{\lambda }^n\left(x\right)-{\gamma }^n\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}+\frac{{\lambda }^{n+1}\left(x\right)-{\gamma }^{n+1}\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}\mathbf{i}}\hfill \\ \hfill & {\displaystyle +\frac{{\lambda }^{n+2}\left(x\right)-{\gamma }^{n+2}\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}\mathsf{\epsilon }+\frac{{\lambda }^{n+3}\left(x\right)-{\gamma }^{n+3}\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}\mathbf{h}}\hfill \\ \hfill & ={\displaystyle \frac{{\lambda }^n\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}}(1+\lambda \left(x\right)\mathbf{i}+{\lambda }^2\left(x\right)\mathsf{\epsilon }+{\lambda }^3\left(x\right)\mathbf{h})\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }^n\left(x\right)}{\lambda \left(x\right)-\gamma \left(x\right)}}(1+\gamma \left(x\right)\mathbf{i}+{\gamma }^2\left(x\right)\mathsf{\epsilon }+{\gamma }^3\left(x\right)\mathbf{h}),\hfill \end{array}$$

which ends the proof. □

Theorem 7.

Let $n\ge 0$ be an integer. Then,

$$\begin{array}{cc}\hfill C{H}_n\left(x\right)& ={\displaystyle \frac{1}{2}}({\lambda }^n\left(x\right)\left(1+\lambda \left(x\right)\mathbf{i}+{\lambda }^2\left(x\right)\mathsf{\epsilon }+{\lambda }^3\left(x\right)\mathbf{h}\right)\hfill \\ \hfill & +{\gamma }^n\left(x\right)\left(1+\gamma \left(x\right)\mathbf{i}+{\gamma }^2\left(x\right)\mathsf{\epsilon }+{\gamma }^3\left(x\right)\mathbf{h}\right)),\hfill \end{array}$$

(12)

where $\lambda \left(x\right)=3x+\sqrt{9x^2-1}$ and $\gamma \left(x\right)={\lambda }^{-1}\left(x\right)=3x-\sqrt{9x^2-1}.$

For the simplicity of notation, let

$$\widehat{\lambda }\left(x\right)=1+\lambda \left(x\right)\mathbf{i}+{\lambda }^2\left(x\right)\mathsf{\epsilon }+{\lambda }^3\left(x\right)\mathbf{h},$$

(13)

$$\widehat{\gamma }\left(x\right)=1+\gamma \left(x\right)\mathbf{i}+{\gamma }^2\left(x\right)\mathsf{\epsilon }+{\gamma }^3\left(x\right)\mathbf{h}.$$

(14)

Moreover,

$$\lambda \left(x\right)-\gamma \left(x\right)=2\sqrt{9x^2-1},$$

$$\lambda \left(x\right)·\gamma \left(x\right)=1.$$

Thus, we can write (11) and (12) as

$$B{H}_n\left(x\right)={\displaystyle \frac{{\lambda }^n\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^n\left(x\right)\widehat{\gamma }\left(x\right)}{2\sqrt{9x^2-1}}},$$

(15)

$$C{H}_n\left(x\right)={\displaystyle \frac{{\lambda }^n\left(x\right)\widehat{\lambda }\left(x\right)+{\gamma }^n\left(x\right)\widehat{\gamma }\left(x\right)}{2}},$$

(16)

respectively.

Theorem 8.

Let $p\ge 0$, $q\ge 0$, $s\ge 0$, $t\ge 0$ be integers such that $p+q=s+t$. Then,

$$\begin{array}{cc}\hfill & B{H}_p\left(x\right)·B{H}_q\left(x\right)-B{H}_s\left(x\right)·B{H}_t\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)({\lambda }^s\left(x\right){\gamma }^t\left(x\right)-{\lambda }^p\left(x\right){\gamma }^q\left(x\right))\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)({\gamma }^s\left(x\right){\lambda }^t\left(x\right)-{\gamma }^p\left(x\right){\lambda }^q\left(x\right))\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are defined in (13) and (14), respectively.

Proof. 

By Formula (15), we obtain

$$\begin{array}{cc}\hfill & B{H}_p\left(x\right)·B{H}_q\left(x\right)-B{H}_s\left(x\right)·B{H}_t\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[({\lambda }^p\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^p\left(x\right)\widehat{\gamma }\left(x\right))({\lambda }^q\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^q\left(x\right)\widehat{\gamma }\left(x\right))\right.\hfill \\ \hfill & \left.-({\lambda }^s\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^s\left(x\right)\widehat{\gamma }\left(x\right))({\lambda }^t\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^t\left(x\right)\widehat{\gamma }\left(x\right))\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[{\lambda }^{p+q}\left(x\right){\left(\widehat{\lambda }\left(x\right)\right)}^2-{\lambda }^p\left(x\right){\gamma }^q\left(x\right)\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\right.\hfill \\ \hfill & -{\gamma }^p\left(x\right){\lambda }^q\left(x\right)\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)+{\gamma }^{p+q}\left(x\right){\left(\widehat{\gamma }\left(x\right)\right)}^2\hfill \\ \hfill & -{\lambda }^{s+t}\left(x\right){\left(\widehat{\lambda }\left(x\right)\right)}^2+{\lambda }^s\left(x\right){\gamma }^t\left(x\right)\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\hfill \\ \hfill & \left.+{\gamma }^s\left(x\right){\lambda }^t\left(x\right)\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^{s+t}\left(x\right){\left(\widehat{\gamma }\left(x\right)\right)}^2\right].\hfill \end{array}$$

Since $p+q=s+t$, we obtain

$$\begin{array}{cc}\hfill & B{H}_p\left(x\right)·B{H}_q\left(x\right)-B{H}_s\left(x\right)·B{H}_t\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)({\lambda }^s\left(x\right){\gamma }^t\left(x\right)-{\lambda }^p\left(x\right){\gamma }^q\left(x\right))\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)({\gamma }^s\left(x\right){\lambda }^t\left(x\right)-{\gamma }^p\left(x\right){\lambda }^q\left(x\right))\right].\hfill \end{array}$$

□

It is easy to see that for special values of $p,q,s,t$, by Theorem 8, we obtain some identities for balancing hybrinomials:

• for $p=n-m$, $q=n+m$, and $s=t=n$ Catalan identity,
• for $p=n-1$, $q=n+1$, and $s=t=n$ Cassini identity,
• for $p=n$, $q=m+1$, $s=n+1$, and $t=m$ d’Ocagne identity,
• for $p=m+k$, $q=n-k$, $s=m$, and $t=n$ Vajda identity.

Corollary 1. 

(Catalan identity for balancing hybrinomials). Let $n\ge 0$, $m\ge 0$ be integers such that $n\ge m$. Then,

$$\begin{array}{cc}\hfill & B{H}_{n-m}\left(x\right)·B{H}_{n+m}\left(x\right)-{\left(B{H}_n\left(x\right)\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\left(1-{\left({\displaystyle \frac{\gamma \left(x\right)}{\lambda \left(x\right)}}\right)}^m\right)+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)\left(1-{\left({\displaystyle \frac{\lambda \left(x\right)}{\gamma \left(x\right)}}\right)}^m\right)\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are defined in (13) and (14), respectively.

Corollary 2 

(Cassini identity for balancing hybrinomials). Let $n\ge 1$ be an integer. Then,

$$\begin{array}{cc}\hfill & B{H}_{n-1}\left(x\right)·B{H}_{n+1}\left(x\right)-{\left(B{H}_n\left(x\right)\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)(2-18x^2+6x\sqrt{9x^2-1})\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)(2-18x^2-6x\sqrt{9x^2-1})\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are defined in (13) and (14), respectively.

Corollary 3 

(d’Ocagne identity for balancing hybrinomials). Let $n\ge 0$, $m\ge 0$ be integers such that $m\ge n$. Then,

$$\begin{array}{cc}\hfill & B{H}_n\left(x\right)·B{H}_{m+1}\left(x\right)-B{H}_{n+1}\left(x\right)·B{H}_m\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\sqrt{9x^2-1}}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right){\gamma }^{m-n}\left(x\right)-\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right){\lambda }^{m-n}\left(x\right)\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are defined in (13) and (14), respectively.

Corollary 4 

(Vajda identity for balancing hybrinomials). Let $n\ge 0$, $m\ge 0$ and $k\ge 0$ be integers such that $n\ge m$ and $n\ge k$. Then,

$$\begin{array}{cc}\hfill & B{H}_{m+k}\left(x\right)·B{H}_{n-k}\left(x\right)-B{H}_m\left(x\right)·B{H}_n\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{36x^2-4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right){\left(\gamma \left(x\right)\right)}^{n-m}\left(1-{\left({\displaystyle \frac{\lambda \left(x\right)}{\gamma \left(x\right)}}\right)}^k\right)\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right){\left(\lambda \left(x\right)\right)}^{n-m}\left(1-{\left({\displaystyle \frac{\gamma \left(x\right)}{\lambda \left(x\right)}}\right)}^k\right)\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are defined in (13) and (14), respectively.

Theorem 9.

Let $p\ge 0$, $q\ge 0$, $s\ge 0$, $t\ge 0$ be integers such that $p+q=s+t$. Then,

$$\begin{array}{cc}\hfill & C{H}_p\left(x\right)·C{H}_q\left(x\right)-C{H}_s\left(x\right)·C{H}_t\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)({\lambda }^p\left(x\right){\gamma }^q\left(x\right)-{\lambda }^s\left(x\right){\gamma }^t\left(x\right))\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)({\gamma }^p\left(x\right){\lambda }^q\left(x\right)-{\gamma }^s\left(x\right){\lambda }^t\left(x\right))\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are defined in (13) and (14), respectively.

Proof. 

By (16), we obtain

$$\begin{array}{cc}\hfill & C{H}_p\left(x\right)·C{H}_q\left(x\right)-C{H}_s\left(x\right)·C{H}_t\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left[{\lambda }^{p+q}\left(x\right){\left(\widehat{\lambda }\left(x\right)\right)}^2+{\lambda }^p\left(x\right){\gamma }^q\left(x\right)\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\right.\hfill \\ \hfill & +{\lambda }^q\left(x\right){\gamma }^p\left(x\right)\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)+{\gamma }^{p+q}\left(x\right){\left(\widehat{\gamma }\left(x\right)\right)}^2\hfill \\ \hfill & -{\lambda }^{s+t}\left(x\right){\left(\widehat{\lambda }\left(x\right)\right)}^2-{\lambda }^s\left(x\right){\gamma }^t\left(x\right)\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\hfill \\ \hfill & \left.-{\lambda }^t\left(x\right){\gamma }^s\left(x\right)\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)-{\gamma }^{s+t}\left(x\right){\left(\widehat{\gamma }\left(x\right)\right)}^2\right].\hfill \end{array}$$

Using the fact that $p+q=s+t$, we obtain the correct result. □

Corollary 5 

(Catalan identity for Lucas-balancing hybrinomials). Let $n\ge 0$, $m\ge 0$ be integers such that $n\ge m$. Then,

$$\begin{array}{cc}\hfill & C{H}_{n-m}\left(x\right)·C{H}_{n+m}\left(x\right)-{\left(C{H}_n\left(x\right)\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\left({\left({\displaystyle \frac{\gamma \left(x\right)}{\lambda \left(x\right)}}\right)}^m-1\right)+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)\left({\left({\displaystyle \frac{\lambda \left(x\right)}{\gamma \left(x\right)}}\right)}^m-1\right)\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are given by (13) and (14), respectively.

Corollary 6 

(Cassini identity for Lucas-balancing hybrinomials). Let $n\ge 1$ be an integer. Then,

$$\begin{array}{cc}\hfill & C{H}_{n-1}\left(x\right)·C{H}_{n+1}\left(x\right)-{\left(C{H}_n\left(x\right)\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right)\left(18x^2-2-6x\sqrt{9x^2-1}\right)\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right)\left(18x^2-2+6x\sqrt{9x^2-1}\right)\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are given by (13) and (14), respectively.

Corollary 7 

(d’Ocagne identity for Lucas-balancing hybrinomials). Let $n\ge 0$, $m\ge 0$ be integers such that $m\ge n$. Then,

$$\begin{array}{cc}\hfill & C{H}_n\left(x\right)·C{H}_{m+1}\left(x\right)-C{H}_{n+1}\left(x\right)·C{H}_m\left(x\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}\sqrt{9x^2-1}\left(\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right){\gamma }^{m-n}\left(x\right)-\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right){\lambda }^{m-n}\left(x\right)\right),\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are given by (13) and (14), respectively.

Corollary 8 

(Vajda identity for Lucas-balancing hybrinomials). Let $n\ge 0$, $m\ge 0$ and $k\ge 0$ be integers such that $n\ge k$ and $n\ge m$. Then,

$$\begin{array}{cc}\hfill & C{H}_{m+k}\left(x\right)·C{H}_{n-k}\left(x\right)-C{H}_m\left(x\right)·C{H}_n\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left[\widehat{\lambda }\left(x\right)\widehat{\gamma }\left(x\right){\left(\gamma \left(x\right)\right)}^{n-m}\left({\left({\displaystyle \frac{\lambda \left(x\right)}{\gamma \left(x\right)}}\right)}^k-1\right)\right.\hfill \\ \hfill & \left.+\widehat{\gamma }\left(x\right)\widehat{\lambda }\left(x\right){\left(\lambda \left(x\right)\right)}^{n-m}\left({\left({\displaystyle \frac{\gamma \left(x\right)}{\lambda \left(x\right)}}\right)}^k-1\right)\right],\hfill \end{array}$$

where $\widehat{\lambda }\left(x\right)$ and $\widehat{\gamma }\left(x\right)$ are given by (13) and (14), respectively.

In the proof of the next theorems, we will use the following results. We omit the proofs of them.

Lemma 1.

Let $n\ge 0$ be an integer. Then,

$$\sum _{l=0}^n{B}_l\left(x\right)={\displaystyle \frac{B_{n+1}\left(x\right)-B_n\left(x\right)-1}{6x-2}},$$

(17)

$$\sum _{l=0}^n{C}_l\left(x\right)={\displaystyle \frac{C_{n+1}\left(x\right)-C_n\left(x\right)+3x-1}{6x-2}}.$$

Theorem 10.

Let $n\ge 0$ be an integer. Then,

$$\sum _{l=0}^{n}B{H}_l\left(x\right)={\displaystyle \frac{B{H}_{n+1}\left(x\right)-B{H}_n\left(x\right)-[1+\mathbf{i}+(6x-1)\mathsf{\epsilon }+(36x^2-6x-1)\mathbf{h}]}{6x-2}}.$$

Proof. 

By (9), we obtain

$$\begin{array}{cc}\hfill & \sum _{l=0}^{n}B{H}_l\left(x\right)=B{H}_0\left(x\right)+B{H}_1\left(x\right)+\dots +B{H}_n\left(x\right)\hfill \\ \hfill & =B_0\left(x\right)+B_1\left(x\right)\mathbf{i}+B_2\left(x\right)\mathsf{\epsilon }+B_3\left(x\right)\mathbf{h}\hfill \\ \hfill & +B_1\left(x\right)+B_2\left(x\right)\mathbf{i}+B_3\left(x\right)\mathsf{\epsilon }+B_4\left(x\right)\mathbf{h}+\dots \hfill \\ \hfill & +B_n\left(x\right)+B_{n+1}\left(x\right)\mathbf{i}+B_{n+2}\left(x\right)\mathsf{\epsilon }+B_{n+3}\left(x\right)\mathbf{h}\hfill \\ \hfill & =B_0\left(x\right)+B_1\left(x\right)+\dots +B_n\left(x\right)\hfill \\ \hfill & +(B_1\left(x\right)+B_2\left(x\right)+\dots +B_{n+1}\left(x\right)+B_0\left(x\right)-B_0\left(x\right))\mathbf{i}\hfill \\ \hfill & +(B_2\left(x\right)+B_3\left(x\right)+\dots +B_{n+2}\left(x\right)+B_0\left(x\right)+B_1\left(x\right)\hfill \\ \hfill & -B_0\left(x\right)-B_1\left(x\right))\mathsf{\epsilon }\hfill \\ \hfill & +(B_3\left(x\right)+B_4\left(x\right)+\dots +B_{n+3}\left(x\right)+B_0\left(x\right)+B_1\left(x\right)+B_2\left(x\right)\hfill \\ \hfill & -B_0\left(x\right)-B_1\left(x\right)-B_2\left(x\right))\mathbf{h}.\hfill \end{array}$$

By (17), we have

$$\sum _{l=0}^n{B}_l\left(x\right)={\displaystyle \frac{B_{n+1}\left(x\right)-B_n\left(x\right)-1}{6x-2}}.$$

Hence, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{l=0}^{n}B{H}_l\left(x\right)=\frac{B_{n+1}\left(x\right)-B_n\left(x\right)-1}{6x-2}}\hfill \\ \hfill & +\left({\displaystyle \frac{B_{n+2}\left(x\right)-B_{n+1}\left(x\right)-1}{6x-2}-B_0\left(x\right)}\right)\mathbf{i}\hfill \\ \hfill & +\left({\displaystyle \frac{B_{n+3}\left(x\right)-B_{n+2}\left(x\right)-1}{6x-2}-B_0\left(x\right)-B_1\left(x\right)}\right)\mathsf{\epsilon }\hfill \\ \hfill & +\left({\displaystyle \frac{B_{n+4}\left(x\right)-B_{n+3}\left(x\right)-1}{6x-2}-B_0\left(x\right)-B_1\left(x\right)-B_2\left(x\right)}\right)\mathbf{h}\hfill \\ \hfill & {\displaystyle =\frac{1}{6x-2}[B_{n+1}\left(x\right)+B_{n+2}\left(x\right)\mathbf{i}+B_{n+3}\left(x\right)\mathsf{\epsilon }+B_{n+4}\left(x\right)\mathbf{h}}\hfill \\ \hfill & -(B_n\left(x\right)+B_{n+1}\left(x\right)\mathbf{i}+B_{n+2}\left(x\right)\mathsf{\epsilon }+B_{n+3}\left(x\right)\mathbf{h})\hfill \\ \hfill & -(1+\mathbf{i}+(6x-1)\mathsf{\epsilon }+(36x^2-6x-1)\mathbf{h})]\hfill \\ \hfill & ={\displaystyle \frac{1}{6x-2}}[B{H}_{n+1}\left(x\right)-B{H}_n\left(x\right)-(1+\mathbf{i}+(6x-1)\mathsf{\epsilon }+(36x^2-6x-1)\mathbf{h})].\hfill \end{array}$$

□

Theorem 11.

Let $n\ge 0$ be an integer. Then,

$$\begin{array}{cc}\hfill & \sum _{l=0}^{n}C{H}_l\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{C{H}_{n+1}\left(x\right)-C{H}_n\left(x\right)+(3x-1)[1-\mathbf{i}-(6x+1)\mathsf{\epsilon }-(36x^2+6x-1)\mathbf{h}]}{6x-2}}.\hfill \end{array}$$

We omit the proof of Theorem 11. It is similar to the proof of Theorem 10.

4. Generating Functions and Matrix Interpretations of Balancing Hybrinomials

Now, we will give the generating function of balancing hybrinomials.

Theorem 12.

The generating function of balancing hybrinomial sequence $\left\{B{H}_n\left(x\right)\right\}$ is

$$G\left(t\right)={\displaystyle \frac{B{H}_0\left(x\right)+(B{H}_1\left(x\right)-6xB{H}_0\left(x\right))t}{1-6xt+t^2}},$$

where $B{H}_0\left(x\right)=\mathbf{i}+6x\mathsf{\epsilon }+(36x^2-1)\mathbf{h}$ and $B{H}_1\left(x\right)-6xB{H}_0\left(x\right)=1-\mathsf{\epsilon }-6x\mathbf{h}.$

Proof. 

Let $G\left(t\right)={\displaystyle \sum _{n=0}^{\infty }}B{H}_n\left(x\right)t^n=B{H}_0\left(x\right)+B{H}_1\left(x\right)t+B{H}_2\left(x\right)t^2+\dots .$ Hence, we obtain

$$-6xtG\left(t\right)=-6xB{H}_0\left(x\right)t-6xB{H}_1\left(x\right)t^2-6xB{H}_2\left(x\right)t^3-\dots$$

$$t^{2}G\left(t\right)=B{H}_0\left(x\right)t^2+B{H}_1\left(x\right)t^3+B{H}_2\left(x\right)t^4+\dots$$

By adding the above equalities, we obtain

$$G\left(t\right)(1-6xt+t^2)=B{H}_0\left(x\right)+(B{H}_1\left(x\right)-6xB{H}_0\left(x\right))t$$

since $B{H}_n\left(x\right)=6B{H}_{n-1}\left(x\right)-B{H}_{n-2}\left(x\right)$ (see (10)) and the coefficients of $t^n$ for $n\ge 2$ are equal to zero. Moreover, by simple calculations, we have

$$B{H}_1\left(x\right)-6xB{H}_0\left(x\right)=1-\mathsf{\epsilon }-6x\mathbf{h}.$$

□

Using the same method, we obtain the generating function of the Lucas-balancing hybrinomial sequence.

Theorem 13.

The generating function of the Lucas-balancing hybrinomial sequence $\left\{C{H}_n\left(x\right)\right\}$ is

$$g\left(t\right)={\displaystyle \frac{C{H}_0\left(x\right)+(C{H}_1\left(x\right)-6xC{H}_0\left(x\right))t}{1-6xt+t^2}},$$

where $C{H}_0\left(x\right)=1+3x\mathbf{i}+(18x^2-1)\mathsf{\epsilon }+(108x^3-9x)\mathbf{h}$

and $C{H}_1\left(x\right)-6xC{H}_0\left(x\right)=-3x-\mathbf{i}-3x\mathsf{\epsilon }+(1-18x^2)\mathbf{h}.$

At the end, we will show a matrix representation of balancing hybrinomials and Lucas-balancing hybrinomials.

Theorem 14.

Let $n\ge 1$ be an integer. Then,

$$\left[\begin{array}{cc}B{H}_{n+1}\left(x\right)\hfill & -B{H}_n\left(x\right)\hfill \\ B{H}_n\left(x\right)\hfill & -B{H}_{n-1}\left(x\right)\hfill \end{array}\right]=\left[\begin{array}{cc}B{H}_2\left(x\right)\hfill & -B{H}_1\left(x\right)\hfill \\ B{H}_1\left(x\right)\hfill & -B{H}_0\left(x\right)\hfill \end{array}\right]·{\left[\begin{array}{cc}\hfill 6x& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]}^{n-1}.$$

(18)

Proof. 

(By the induction on n) If $n=1$, then the result holds. Assuming the Formula (18) is true for $n\ge 1$, we will prove it for $n+1$. By the induction’s hypothesis and Formula (10), we obtain

$$\left[\begin{array}{cc}B{H}_2\left(x\right)\hfill & -B{H}_1\left(x\right)\hfill \\ B{H}_1\left(x\right)\hfill & -B{H}_0\left(x\right)\hfill \end{array}\right]·{\left[\begin{array}{cc}\hfill 6x& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]}^n$$

$$=\left[\begin{array}{cc}B{H}_{n+1}\left(x\right)\hfill & -B{H}_n\left(x\right)\hfill \\ B{H}_n\left(x\right)\hfill & -B{H}_{n-1}\left(x\right)\hfill \end{array}\right]·\left[\begin{array}{cc}\hfill 6x& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]$$

$$=\left[\begin{array}{cc}6xB{H}_{n+1}\left(x\right)-B{H}_n\left(x\right)\hfill & -B{H}_{n+1}\left(x\right)\hfill \\ 6xB{H}_n\left(x\right)-B{H}_{n-1}\left(x\right)\hfill & -B{H}_n\left(x\right)\hfill \end{array}\right]$$

$$=\left[\begin{array}{cc}B{H}_{n+2}\left(x\right)\hfill & -B{H}_{n+1}\left(x\right)\hfill \\ B{H}_{n+1}\left(x\right)\hfill & -B{H}_n\left(x\right)\hfill \end{array}\right],$$

which completes the proof. □

In the same way, we can prove the following result for Lucas-balancing hybrinomials.

Theorem 15.

Let $n\ge 1$ be an integer. Then,

$$\left[\begin{array}{cc}C{H}_{n+1}\left(x\right)\hfill & -C{H}_n\left(x\right)\hfill \\ C{H}_n\left(x\right)\hfill & -C{H}_{n-1}\left(x\right)\hfill \end{array}\right]=\left[\begin{array}{cc}C{H}_2\left(x\right)\hfill & -C{H}_1\left(x\right)\hfill \\ C{H}_1\left(x\right)\hfill & -C{H}_0\left(x\right)\hfill \end{array}\right]·{\left[\begin{array}{cc}\hfill 6x& \hfill -1\\ \hfill 1& \hfill 0\end{array}\right]}^{n-1}.$$

5. Conclusions

In this paper, we introduced and studied balancing and Lucas-balancing hybrinomials. It is known (see [28]) that balancing and Lucas-balancing polynomials are “rescaled” Chebyshev polynomials. Balancing and Lucas-balancing polynomials can be also expressed in terms of Legendre polynomials. In future work, the researchers can define and investigate Chebyshev and Legendre hybrinomials. It will be interesting to continue this research by examining the connections between these hybrinomials.

The papers [37,38] concern some identities and relationships between balancing type polynomials. Based on these concepts, it is natural to consider new properties of balancing type hybrinomials as a future research direction.