Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions

Abstract

We have investigated new Pauli Fibonacci and Pauli Lucas quaternions by taking the components of these quaternions as Gaussian Fibonacci and Gaussian Lucas numbers, respectively. We have calculated some basic identities for these quaternions. Later, the generating functions and Binet formulas are obtained for Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions. Furthermore, Honsberger’s identity, Catalan’s and Cassini’s identities have been given for Pauli Gaussian Fibonacci quaternions.

1. Introduction

The set of complex numbers with integer coefficients was first described by Carl Friedrich Gauss, and these numbers were called Gaussiannumbers [1]. Then, Horadam gave the definition of the n-th generalized complex Fibonacci quaternion and provided some identities regarding these numbers. Furthermore, he defined Fibonacci quaternions [2]. Gaussian Fibonacci and Gaussian Lucas sequences were introduced by Jordan. Moreover, some basic identities and summation formulas were obtained [3]. The recurrence relation of the Gaussian Fibonacci numbers $G{F}_n$ for $n>1$ is defined by

$$G{F}_n=G{F}_{n-1}+G{F}_{n-2}$$

where $G{F}_0=i,G{F}_1=1$. These recurrence relations also satisfy the following equality $G{F}_n=F_n+i{F}_{n-1}$ where $F_n$ is the n-th Fibonacci number [3,4]. In the same manner, the recurrence relation of the Gaussian Lucas numbers $G{L}_n$ for $n>2$ is defined by

$$G{L}_n=G{L}_{n-1}+G{L}_{n-2}$$

where $G{L}_0=2-i,G{L}_1=1+2i$. Again, the following equality can be observed

$$G{L}_n=L_n+i{L}_{n-1}$$

where $L_n$ is the n-th Lucas number [3,4].

An extension of the Fibonacci numbers to the complex plane was discussed by Berzsenyi [5].

The algebras of the complex numbers, quaternions, octonions, and sedenions are found by using a doubling procedure. This procedure is called as Cayley–Dickson process. In this regard, we extend the field of real numbers to complex numbers via this process. The complex number system is both commutative and associative. However, the quaternions are not commutative, although they are associative. On the other hand, octonions and sedenions are both non-commutative and non-associative. The main question is, why do we need these expanding number systems? It is because quaternions have applications in quantum mechanics, computer graphics, and vision [6,7,8]. Octonions are used in quantum information theory and robotics [9,10]. Sedenions are used in neural networks, time series, and traffic forecasting problems [11,12].

The sequences in finite fields whose terms depend in a simple manner on their predecessors are of importance for a variety of applications. Because it is easy to generate by recursive procedures, and these sequences have advantageous features from the computational viewpoint [13]. Thus, mathematicians increased the number of terms to be added at the beginning and turned to studies on number sequences similar to Fibonacci numbers, such as tribonacci, tetranacci, pentanacci, etc. Later, these studies were carried over to Cayley algebras, see [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. Thus, one of the most active research areas of recent years has come to the fore, and studies on Cayley algebras have attracted researchers in various ways.

Complex Fibonacci quaternions have been defined by Halici [33]. Then, Binet’s formula, generating functions and the matrix representation of these quaternions have been proven. Recently, n-th quaternion Gaussian Lucas numbers have been introduced to the literature. The Binet formula, some summation formulas, and the Cassini identity have been given by using the matrix representation of Gaussian Lucas numbers [34]. This time, they have expressed the quaternions instead of the Gaussian Lucas numbers by taking the Gaussian Fibonacci coefficients. In this way, the Binet formula, generating function ve some identities regarding the norm of these quaternions have been derived [35].

The Binet formula of the Gaussian Fibonacci sequence and Gaussian Lucas sequence are

$$G{F}_n=\frac{1}{\alpha -\beta }\left\{\left(1-i\beta \right){\alpha }^n-\left(1-i\alpha \right){\beta }^n\right\}$$

and

$$G{L}_n=\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }G{L}_1+\frac{{\alpha }^{n-1}-{\beta }^{n-1}}{\alpha -\beta }G{L}_0$$

respectively, where $\alpha$ and $\beta$ denote the roots of the characteristic equation for Gaussian Fibonacci sequence and $G{L}_0,G{L}_1$ denote the initial values for the Gaussian Lucas numbers [34,35].

The Pauli matrices that have been introduced by Wolfgang Pauli form a set of $2\times 2$ complex matrices as follows:

$$1=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],{\sigma }_1=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_2=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\sigma }_3=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

The multiplication rules are given by

$$\begin{array}{c}{\sigma }_1^2={\sigma }_2^2={\sigma }_3^2=1,{\sigma }_1{\sigma }_2=-{\sigma }_2{\sigma }_1=i{\sigma }_3\hfill \\ {\sigma }_2{\sigma }_3=-{\sigma }_3{\sigma }_2=i{\sigma }_1,{\sigma }_3{\sigma }_1=-{\sigma }_1{\sigma }_3=i{\sigma }_2\hfill \end{array}$$

Further, these matrices are Hermitian and unitary. These $2\times 2$ types of Hermitian matrices form a basis for the real vector space and the span of $\left\{I,i{\sigma }_1,i{\sigma }_2,i{\sigma }_3\right\}$ is isomorphic to the real algebra of quaternions [36,37].

The Pauli quaternions are defined by Kim as follows:

$$q=a_{0}1+a_1{\sigma }_1+a_2{\sigma }_2+a_3{\sigma }_3,$$

where $1,{\sigma }_1,{\sigma }_2$ and ${\sigma }_3$ represent the Pauli matrices.

Let $q=a_{0}1+a_1{\sigma }_1+a_2{\sigma }_2+a_3{\sigma }_3$ and $p=b_{0}1+b_1{\sigma }_1+b_2{\sigma }_2+b_3{\sigma }_3$ be Pauli quaternions, then the product of these quaternions are given by [37]

$$\begin{array}{c}q.p=\left(a_0{b}_0+a_1{b}_1+a_2{b}_2+a_3{b}_3\right)1+\left\{\left(a_0{b}_1+a_1{b}_0\right)+i\left(a_2{b}_3-a_3{b}_2\right)\right\}{\sigma }_1\hfill \\ +\left\{\left(a_0{b}_2+a_2{b}_0\right)+i\left(a_3{b}_1-a_1{b}_3\right)\right\}{\sigma }_2\hfill \\ +\left\{\left(a_0{b}_3+a_3{b}_0\right)+i\left(a_1{b}_2-a_2{b}_1\right)\right\}{\sigma }_3.\hfill \end{array}$$

The conjugate and the norm of Pauli quaternions are

$$q^{*}=a_{0}1-a_1{\sigma }_1-a_2{\sigma }_2-a_3{\sigma }_3$$

and

$$N\left(q\right)=\sqrt{\left|q.q^{*}\right|}=\sqrt{\left|a_0^2-a_1^2-a_2^2-a_3^2\right|},$$

respectively [37].

Torunbalcı has presented Pauli Fibonacci and Pauli Lucas quaternions by taking the real coefficients of Pauli quaternion as the Fibonacci number sequence. Honsberger’s, d’Ocagne’s, Catalan, Cassini identities, generating function and Binet formula and the matrix representation have been given for the Pauli Fibonacci quaternions [38].

In a recent paper [39] on a base of quaternions, the families of associated sequences of real polynomials and numbers were defined. Quaternion equivalents for quasi-Fibonacci numbers (shortly quaternaccis) were introduced. In OEIS, there is a number of quaternacci sequences connected with generalized Gaussian Fibonacci integers. We are also interested in quaternions with coefficients of Gaussian Fibonacci numbers.

Especially, in this work, our aim is to introduce Pauli Fibonacci quaternions and Pauli Lucas quaternions whose coefficients consist of Gaussian Fibonacci numbers and Gaussian Lucas numbers, respectively. We called these numbers the Pauli Gaussian Fibonacci and Pauli Gaussian Lucas numbers, respectively. Then, some algebraic properties of these quaternions have been shown. Moreover, some identities and formulas for these quaternions have been obtained.

2. The Pauli Gaussian Fibonacci Quaternions

In this section, the definition of Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions will be given. Then, some algebraic properties, identities and theorems are given for Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions.

Definition 1. 

The Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions are defined by

$$Q_{p}G{F}_n=G{F}_{n}1+G{F}_{n+1}{\sigma }_1+G{F}_{n+2}{\sigma }_2+G{F}_{n+3}{\sigma }_3$$

and

$$Q_{p}G{L}_n=G{L}_{n}1+G{L}_{n+1}{\sigma }_1+G{L}_{n+2}{\sigma }_2+G{L}_{n+3}{\sigma }_3$$

respectively, where $G{F}_n$ and $G{L}_n$ are the n-th Gaussian Fibonacci and Gaussian Lucas numbers.

Furthermore, these numbers are related to Pauli Fibonacci and Pauli Lucas quaternions as follows

$$Q_{p}G{F}_n=Q_p{F}_n+i{Q}_p{F}_{n-1}$$

and

$$Q_{p}G{L}_n=Q_p{L}_n+i{Q}_p{L}_{n-1}$$

respectively.

In order to obtain the recursive relations for Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions, we will consider the relations $G{F}_{n+2}=G{F}_n+G{F}_{n+1}$ and $G{L}_{n+2}=G{L}_n+G{L}_{n+1}$ for Gaussian Fibonacci and Gaussian Lucas numbers, respectively.

Hence, for $n\ge 0$

$$Q_{p}G{F}_{n+2}=Q_{p}G{F}_n+Q_{p}G{F}_{n+1}$$

and

$$Q_{p}G{L}_{n+2}=Q_{p}G{L}_n+Q_{p}G{L}_{n+1}$$

respectively.

Definition 2. 

The conjugates of the Pauli Gaussian Fibonacci quaternion $Q_{p}G{F}_n$ and the Pauli Gaussian Lucas quaternion $Q_{p}G{L}_n$ are defined by

$$\overline{Q_{p}G{F}_n}=G{F}_{n}1-G{F}_{n+1}{\sigma }_1-G{F}_{n+2}{\sigma }_2-G{F}_{n+3}{\sigma }_3$$

and

$$\overline{Q_{p}G{L}_n}=G{L}_{n}1-G{L}_{n+1}{\sigma }_1-G{L}_{n+2}{\sigma }_2-G{L}_{n+3}{\sigma }_3$$

respectively.

The addition, subtraction and multiplication of two Pauli Gaussian Fibonacci quaternions $Q_{p}G{F}_n$ and $Q_{p}G{F}_m$ aregiven by

$$\begin{array}{c}{Q}_{p}G{F}_n\pm Q_{p}G{F}_m=\left(G{F}_n\pm G{F}_m\right).1+\left(G{F}_{n+1}\pm G{F}_{m+1}\right){\sigma }_1\hfill \\ +\left(G{F}_{n+2}\pm G{F}_{m+2}\right){\sigma }_2+\left(G{F}_{n+3}\pm G{F}_{m+3}\right){\sigma }_3\hfill \end{array}$$

(1)

and

$$\begin{array}{c}{Q}_{p}G{F}_n\times Q_{p}G{F}_m=\left(G{F}_n.G{F}_m+G{F}_{n+1}.G{F}_{m+1}+G{F}_{n+2}.G{F}_{m+2}+G{F}_{n+3}.G{F}_{m+3}\right).1\hfill \\ +\left(G{F}_{n+1}.G{F}_m+G{F}_n.G{F}_{m+1}-iG{F}_{n+3}.G{F}_{m+2}+iG{F}_{n+2}.G{F}_{m+3}\right){\sigma }_1\hfill \\ +\left(G{F}_{n+2}.G{F}_m+iG{F}_{n+3}.G{F}_{m+1}+G{F}_n.G{F}_{m+2}-iG{F}_{n+1}.G{F}_{m+3}\right){\sigma }_2\hfill \\ +\left(G{F}_{n+3}.G{F}_m-iG{F}_{n+2}.G{F}_{m+1}+iG{F}_{n+1}.G{F}_{m+2}+G{F}_n.G{F}_{m+3}\right){\sigma }_3\hfill \end{array}$$

(2)

respectively.

Note that $Q_{p}G{F}_n\times Q_{p}G{F}_m\ne Q_{p}G{F}_m\times Q_{p}G{F}_n.$

In this case, the norm of any Pauli Gaussian Fibonacci quaternion can be written as

$$N_{Q_{p}G{F}_n}^2=Q_{p}G{F}_n\times \overline{Q_{p}G{F}_n}=\left|G{F}_n^2-G{F}_{n+1}^2-G{F}_{n+2}^2-G{F}_{n+3}^2\right|.$$

So, the scalar and vectorial part of any Pauli Gaussian Fibonacci quaternion is represented by

$$S_{Q_{p}G{F}_n}=G{F}_n,V_{Q_{p}G{F}_n}=G{F}_{n+1}{\sigma }_1+G{F}_{n+2}{\sigma }_2+G{F}_{n+3}{\sigma }_3.$$

In addition, Equation (2) can be rewritten in terms of the scalar and vector parts of the Pauli Gaussian Fibonacci quaternion as follows.

$$\begin{array}{c}{Q}_{p}G{F}_n\times Q_{p}G{F}_m=S_{Q_{p}G{F}_n}{S}_{Q_{p}G{F}_m}+⟨V_{Q_{p}G{F}_n},V_{Q_{p}G{F}_m}⟩+S_{Q_{p}G{F}_n}{V}_{Q_{p}G{F}_m}\hfill \\ +S_{Q_{p}G{F}_m}{V}_{Q_{p}G{F}_n}+V_{Q_{p}G{F}_n}\wedge V_{Q_{p}G{F}_m}.\hfill \end{array}$$

With the aid of Equation (2), the following Pauli Gaussian Fibonacci quaternion can be expressed as a matrix form

$$Q_{p}G{F}_n\times Q_{p}G{F}_m=\left[\begin{array}{cccc}G{F}_n& G{F}_{n+1}& G{F}_{n+2}& G{F}_{n+3}\\ G{F}_{n+1}& G{F}_n& -iG{F}_{n+3}& iG{F}_{n+2}\\ G{F}_{n+2}& iG{F}_{n+3}& G{F}_n& -iG{F}_{n+1}\\ G{F}_{n+3}& -iG{F}_{n+2}& iG{F}_{n+1}& G{F}_n\end{array}\right]\left[\begin{array}{c}G{F}_m\\ G{F}_{m+1}\\ G{F}_{m+2}\\ G{F}_{m+3}\end{array}\right].$$

Theorem 1. 

Let $Q_{p}G{F}_n$, $Q_{p}G{L}_n$ and $G{F}_n$ denote the Gaussian Fibonacci number, Gaussian Lucas number and the Gaussian Fibonacci number, respectively. For $n\ge 1$, we get the following relations

(i)

$$Q_{p}G{F}_{n+1}+Q_{p}G{F}_{n-1}=Q_{p}G{L}_n$$

(ii)

$$Q_{p}G{F}_n+Q_{p}G{F}_{n-1}=Q_{p}G{F}_{n+1}$$

(iii)

$$Q_{p}G{F}_{n+2}-Q_{p}G{F}_{n-2}=Q_{p}G{L}_n$$

(iv)

$$Q_{p}G{F}_n-Q_{p}G{F}_{n+1}{\sigma }_1-Q_{p}G{F}_{n+2}{\sigma }_2-Q_{p}G{F}_{n+3}{\sigma }_3=G{F}_n-G{F}_{n+2}-G{F}_{n+4}-G{F}_{n+6}$$

Proof. 

(i) Considering Equation (1) and using the identity $G{F}_{n+1}+G{F}_{n-1}=G{L}_n$ [3], we have the proof as follows

$$\begin{array}{c}{Q}_{p}G{F}_{n+1}+Q_{p}G{F}_{n-1}=\left(G{F}_{n+1}+G{F}_{n-1}\right).1+\left(G{F}_{n+2}+G{F}_n\right){\sigma }_1\hfill \\ +\left(G{F}_{n+3}+G{F}_{n+1}\right){\sigma }_2+\left(G{F}_{n+4}+G{F}_{n+2}\right){\sigma }_3\hfill \\ =G{L}_{n}1+G{L}_{n+1}{\sigma }_1+G{L}_{n+2}{\sigma }_2+G{L}_{n+3}{\sigma }_3\hfill \\ =Q_{p}G{L}_n.\hfill \end{array}$$

(ii) 

If we use Equation (1) and the recurrence relation of the Gaussian Fibonacci numbers, the proof can be easily seen.

(iii) 

Using Equation (1) and the recurrence relation of the Gaussian Fibonacci numbers, we obtain

$$\begin{array}{c}{Q}_{p}G{F}_{n+2}-Q_{p}G{F}_{n-2}=\left(G{F}_{n+1}+G{F}_{n-1}\right).1+\left(G{F}_{n+2}+G{F}_n\right){\sigma }_1\hfill \\ +\left(G{F}_{n+3}+G{F}_{n+1}\right){\sigma }_2+\left(G{F}_{n+4}+G{F}_{n+2}\right){\sigma }_3.\hfill \end{array}$$

By substituting the identity $G{F}_{n+1}+G{F}_{n-1}=G{L}_n$ [3] into the previous equation we get

$$Q_{p}G{F}_{n+2}-Q_{p}G{F}_{n-2}=Q_{p}G{L}_n.$$

(iv) 

Multiplying both sides of the Pauli Gaussian Fibonacci quaternions $Q_{p}G{F}_{n+1},Q_{p}G{F}_{n+2}$, $Q_{p}G{F}_{n+3}$ by $-{\sigma }_1,-{\sigma }_2$ and $-{\sigma }_3$ respectively gives

$$\begin{array}{c}-Q_{p}G{F}_{n+1}{\sigma }_1-Q_{p}G{F}_{n+2}{\sigma }_2-Q_{p}G{F}_{n+3}{\sigma }_3=-G{F}_{n+1}{\sigma }_1-G{F}_{n+2}1+iG{F}_{n+3}{\sigma }_3-iG{F}_{n+4}{\sigma }_2\hfill \\ -G{F}_{n+2}{\sigma }_2-iG{F}_{n+3}{\sigma }_3-G{F}_{n+4}1+iG{F}_{n+5}{\sigma }_1\hfill \\ -G{F}_{n+3}{\sigma }_3+iG{F}_{n+4}{\sigma }_2-iG{F}_{n+5}{\sigma }_1-G{F}_{n+6}1\hfill \\ =-G{F}_{n+1}{\sigma }_1-G{F}_{n+2}{\sigma }_2-G{F}_{n+3}{\sigma }_3\hfill \\ -\left(G{F}_{n+2}+G{F}_{n+4}+G{F}_{n+6}\right)1.\hfill \end{array}$$

Then, adding the above equation with $Q_{p}G{F}_n$ yields

$$\begin{array}{c}{Q}_{p}G{F}_n-Q_{p}G{F}_{n+1}{\sigma }_1-Q_{p}G{F}_{n+2}{\sigma }_2-Q_{p}G{F}_{n+3}{\sigma }_3\hfill \\ =G{F}_{n}1+G{F}_{n+1}{\sigma }_1+G{F}_{n+2}{\sigma }_2+G{F}_{n+3}{\sigma }_3-G{F}_{n+1}{\sigma }_1-G{F}_{n+2}{\sigma }_2-G{F}_{n+3}{\sigma }_3\hfill \\ -\left(G{F}_{n+2}+G{F}_{n+4}+G{F}_{n+6}\right)1\hfill \\ =\left(G{F}_n-G{F}_{n+2}-G{F}_{n+4}-G{F}_{n+6}\right)1.\hfill \end{array}$$

□

Theorem 2 

(Honsberger’s Identity). For $n,m\ge 0$ and $G{F}_n$, the Honsberger identity for the Pauli Gaussian Fibonacci quaternions is given by

$$Q_{p}G{F}_n\times Q_{p}G{F}_m+Q_{p}G{F}_{n+1}\times Q_{p}G{F}_{m+1}=\left(2Q_{p}G{F}_{n+m}+9F_{n+m+1}+5F_{n+m+2}\right)\left(1+2i\right).$$

Proof. 

By the Equations (1) and (2), we get

$$\begin{array}{c}{Q}_{p}G{F}_n\times Q_{p}G{F}_m+Q_{p}G{F}_{n+1}\times Q_{p}G{F}_{m+1}\hfill \\ =\left[\left(G{F}_n.G{F}_m+G{F}_{n+1}.G{F}_{m+1}\right)+\left(G{F}_{n+2}.G{F}_{m+2}+G{F}_{n+3}.G{F}_{m+3}\right)\right.\hfill \\ \left.+\left(G{F}_{n+1}.G{F}_{m+1}+G{F}_{n+2}.G{F}_{m+2}\right)+\left(G{F}_{n+3}.G{F}_{m+3}+G{F}_{n+4}.G{F}_{m+4}\right)\right]1\hfill \\ +\left[\left(G{F}_{n+1}.G{F}_m+G{F}_{n+2}.G{F}_{m+1}\right)+\left(G{F}_n.G{F}_{m+1}+G{F}_{n+1}.G{F}_{m+2}\right)\right.\hfill \\ \left.+i\left(G{F}_{n+2}.G{F}_{m+3}+G{F}_{n+3}.G{F}_{m+4}\right)-i\left(G{F}_{n+3}.G{F}_{m+2}+G{F}_{n+4}.G{F}_{m+3}\right)\right]{\sigma }_1\hfill \\ +\left[\left(G{F}_{n+2}.G{F}_m+G{F}_{n+3}.G{F}_{m+1}\right)+\left(G{F}_n.G{F}_{m+2}+G{F}_{n+1}.G{F}_{m+3}\right)\right.\hfill \\ \left.+i\left(G{F}_{n+3}.G{F}_{m+1}+G{F}_{n+4}.G{F}_{m+2}\right)-i\left(G{F}_{n+1}.G{F}_{m+3}+G{F}_{n+2}.G{F}_{m+4}\right)\right]{\sigma }_2\hfill \\ +\left[\left(G{F}_{n+3}.G{F}_m+G{F}_{n+4}.G{F}_{m+1}\right)+\left(G{F}_n.G{F}_{m+3}+G{F}_{n+1}.G{F}_{m+4}\right)\right.\hfill \\ \left.+i\left(G{F}_{n+1}.G{F}_{m+2}+G{F}_{n+2}.G{F}_{m+3}\right)-i\left(G{F}_{n+2}.G{F}_{m+1}+G{F}_{n+3}.G{F}_{m+2}\right)\right]{\sigma }_3.\hfill \end{array}$$

Using the identity $G{F}_{n}G{F}_m+G{F}_{n+1}G{F}_{m+1}=F_{n+m}\left(1+2i\right)$ [3], we obtain

$$\begin{array}{c}{Q}_{p}G{F}_n\times Q_{p}G{F}_m+Q_{p}G{F}_{n+1}\times Q_{p}G{F}_{m+1}\hfill \\ =\left(\left(F_{n+m}+F_{n+m+2}+F_{n+m+4}+F_{n+m+6}\right).1\right.\hfill \\ \left.+2\left(F_{n+m+1}{\sigma }_1+F_{n+m+2}{\sigma }_2+F_{n+m+3}{\sigma }_3\right)\right)\left(1+2i\right).\hfill \end{array}$$

If the necessary arrangements are made in the last equation, we have

$$Q_{p}G{F}_n\times Q_{p}G{F}_m+Q_{p}G{F}_{n+1}\times Q_{p}G{F}_{m+1}=\left(2Q_{p}G{F}_{n+m}+9F_{n+m+1}1+5F_{n+m+2}1\right)\left(1+2i\right).$$

Thus, the claim is verified. □

Theorem 3 

(Generating Function). The generating functions of the Pauli Gaussian Fibonacci quaternions and Pauli Gaussian Lucas quaternions are as follows:

$$g\left(x\right)=\sum _{n=0}^{\infty }{Q}_{p}G{F}_n.x^n=\frac{Q_{p}G{F}_0+\left(Q_{p}G{F}_1-Q_{p}G{F}_0\right)x}{1-x-x^2}$$

and

$$h\left(x\right)=\sum _{n=0}^{\infty }{Q}_{p}G{L}_n.x^n=\frac{Q_{p}G{L}_0+\left(Q_{p}G{L}_1-Q_{p}G{L}_0\right)x}{1-x-x^2}$$

respectively.

Proof. 

Let us use the definition of a generating function of $Q_{p}G{F}_n$ as follows

$$g\left(x\right)=Q_{p}G{F}_0+Q_{p}G{F}_1.x+Q_{p}G{F}_2.x^2+\cdots +Q_{p}G{F}_n.x^n+\cdots$$

(3)

Multiplying both sides of the Equation (3) by $-x$ and $-x^2$ gives

$$-xg\left(x\right)=-Q_{p}G{F}_0.x-Q_{p}G{F}_1.x^2-Q_{p}G{F}_2.x^3-\cdots +Q_{p}G{F}_n.x^{n+1}-\cdots$$

(4)

and

$$-x^{2}g\left(x\right)=-Q_{p}G{F}_0.x^2-Q_{p}G{F}_1.x^3-Q_{p}G{F}_2.x^4-\cdots +Q_{p}G{F}_n.x^{n+2}-\cdots$$

(5)

If we add the Equations (3)–(5) and use Theorem 1, we conclude that

$$\left(1-x-x^2\right)g\left(x\right)=Q_{p}G{F}_0+\left(Q_{p}G{F}_1-Q_{p}G{F}_0\right)x$$

Then, we write

$$g\left(x\right)=\sum _{n=0}^{\infty }{Q}_{p}G{F}_n.x^n=\frac{Q_{p}G{F}_0+\left(Q_{p}G{F}_1-Q_{p}G{F}_0\right)x}{1-x-x^2}.$$

Let us write the generating function of $Q_{p}G{L}_n$ as follows

$$h\left(x\right)=Q_{p}G{L}_0+Q_{p}G{L}_1.x+Q_{p}G{L}_2.x^2+\cdots +Q_{p}G{L}_n.x^n+\cdots$$

(6)

The proof can be easily seen if we apply a similar method used to prove the generating function for the Pauli Gaussian Fibonacci quaternions to the Equation (6).  □

Now, we will obtain the Binet formulas, which give us the n-th Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions, respectively.

Theorem 4 

(Binet’s Formula). (i) For $n\ge 1$, Binet formula of the Pauli Gaussian Fibonacci quaternions is given by

$$Q_{p}G{F}_n=c{\alpha }^n\widehat{\alpha }+d{\beta }^n\widehat{\beta }$$

where $c=\frac{1-\beta i}{\alpha -\beta }$, $d=\frac{-1+\alpha i}{\alpha -\beta }$, $\alpha =\frac{1+\sqrt{5}}{2}$ and $\beta =\frac{1-\sqrt{5}}{2}.$

(ii)  For $n\in \mathrm{N}$, the Binet formula of the Pauli Gaussian Lucas quaternions is given by

$$Q_{p}G{L}_n=Q_P{F}_{n}G{L}_1+Q_P{F}_{n-1}G{L}_0.$$

This last formula gives us the relationship between Pauli Fibonacci quaternions and Gaussian Lucas numbers.

Proof. 

(i) Applying the Binet’s formula of the Gaussian Fibonacci to $Q_{p}G{F}_n$, we get

$$\begin{array}{c}{Q}_{p}G{F}_n=\left(c{\alpha }^n\widehat{\alpha }+d{\beta }^n\widehat{\beta }\right)1+\left(c{\alpha }^{n+1}\widehat{\alpha }+d{\beta }^{n+1}\widehat{\beta }\right){\sigma }_1\hfill \\ +\left(c{\alpha }^{n+2}\widehat{\alpha }+d{\beta }^{n+2}\widehat{\beta }\right){\sigma }_2+\left(c{\alpha }^{n+3}\widehat{\alpha }+d{\beta }^{n+3}\widehat{\beta }\right){\sigma }_3.\hfill \end{array}$$

(7)

If Equation (7) is arranged, we have

$$\begin{array}{c}{Q}_{p}G{F}_n=c{\alpha }^n\left(1+\alpha {\sigma }_1+{\alpha }^2{\sigma }_2+{\alpha }^3{\sigma }_3\right)+d{\beta }^n\left(1+\beta {\sigma }_1+{\beta }^2{\sigma }_2+{\beta }^3{\sigma }_3\right)\hfill \\ Q_{p}G{F}_n=c{\alpha }^n\widehat{\alpha }+d{\beta }^n\widehat{\beta }.\hfill \end{array}$$

such that

$$\widehat{\alpha }=1+\alpha {\sigma }_1+{\alpha }^2{\sigma }_2+{\alpha }^3{\sigma }_3$$

and

$$\widehat{\beta }=1+\beta {\sigma }_1+{\beta }^2{\sigma }_2+{\beta }^3{\sigma }_3.$$

(ii) Applying Binet’s formula of the Gaussian Lucas to $Q_{p}G{L}_n$, we get

$$\begin{array}{c}{Q}_{p}G{L}_n=\left(\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }G{L}_1+\frac{{\alpha }^{n-1}-{\beta }^{n-1}}{\alpha -\beta }G{L}_0\right)1\hfill \\ +\left(\frac{{\alpha }^{n+1}-{\beta }^{n+1}}{\alpha -\beta }G{L}_1+\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }G{L}_0\right){\sigma }_1\hfill \\ +\left(\frac{{\alpha }^{n+2}-{\beta }^{n+2}}{\alpha -\beta }G{L}_1+\frac{{\alpha }^{n+1}-{\beta }^{n+1}}{\alpha -\beta }G{L}_0\right){\sigma }_2\hfill \\ +\left(\frac{{\alpha }^{n+3}-{\beta }^{n+3}}{\alpha -\beta }G{L}_1+\frac{{\alpha }^{n+2}-{\beta }^{n+2}}{\alpha -\beta }G{L}_0\right){\sigma }_3.\hfill \end{array}$$

(8)

Equation (8) can be stated in terms of Fibonacci numbers as follows:

$$\begin{array}{c}{Q}_{p}G{L}_n=\left(F_{n}1+F_{n+1}{\sigma }_1+F_{n+2}{\sigma }_2+F_{n+3}{\sigma }_3\right)G{L}_1\hfill \\ +\left(F_{n-1}1+F_n{\sigma }_1+F_{n+1}{\sigma }_2+F_{n+2}{\sigma }_3\right)G{L}_0\hfill \\ =Q_p{F}_{n}G{L}_1+Q_p{F}_{n-1}G{L}_0.\hfill \end{array}$$

□

Example 1. 

Let $Q_{p}G{F}_2$ be Pauli Gaussian Fibonacci quaternion. Applying Theorem 4 for $n=2$, we get

$$\begin{array}{c}{Q}_{p}G{F}_2=\frac{\left(1-\beta i\right){\alpha }^2\left(1+\alpha {\sigma }_1+{\alpha }^2{\sigma }_2+{\alpha }^3{\sigma }_3\right)+\left(-1+\alpha i\right){\beta }^2\left(1+\beta {\sigma }_1+{\beta }^2{\sigma }_2+{\beta }^3{\sigma }_3\right)}{\sqrt{5}}\hfill \\ =\left(1+i\right)1+\left(2+i\right){\sigma }_1+\left(3+2i\right){\sigma }_2+\left(5+3i\right){\sigma }_3.\hfill \end{array}$$

Furthermore, the above Pauli Gaussian Fibonacci quaternion is written by

$$\begin{array}{c}{Q}_{p}G{F}_2=\left(11+2{\sigma }_1+3{\sigma }_2+5{\sigma }_3\right)+i\left(11+1{\sigma }_1+2{\sigma }_2+3{\sigma }_3\right)\hfill \\ Q_{p}G{F}_2=Q_p{F}_2+i{Q}_p{F}_1.\hfill \end{array}$$

Notice that the real and the imaginary parts of the Pauli Gaussian Fibonacci quaternion correspond Pauli Fibonacci quaternions for $n=2$ and $n=1$.

Theorem 5 

(d’Ocagne’s Identity). For $n,m\ge 0$, the following identity holds

$$Q_{p}G{F}_m\times Q_{p}G{F}_{n+1}-Q_{p}G{F}_{m+1}\times Q_{p}G{F}_n=\left(\frac{i-2}{\sqrt{5}}\right)\left[{\beta }^m{\alpha }^n\widehat{\beta }\widehat{\alpha }-{\beta }^m{\alpha }^n\widehat{\alpha }\widehat{\beta }\right].$$

Proof. 

Considering the Binet formula in Theorem 4 and making some necessary calculations, the following expression is obtained.

$$\begin{array}{c}{Q}_{p}G{F}_m\times Q_{p}G{F}_{n+1}-Q_{p}G{F}_{m+1}\times Q_{p}G{F}_n\hfill \\ =\left(cd\right)\left[\left({\beta }^m{\alpha }^{n+1}-{\beta }^{m+1}{\alpha }^n\right)\widehat{\beta }\widehat{\alpha }+\left({\alpha }^m{\beta }^{n+1}-{\alpha }^{m+1}{\beta }^n\right)\widehat{\alpha }\widehat{\beta }\right]\hfill \\ =\left(cd\right)\left[{\beta }^m{\alpha }^n\left(\alpha -\beta \right)\widehat{\beta }\widehat{\alpha }-{\beta }^m{\alpha }^n\left(\alpha -\beta \right)\widehat{\alpha }\widehat{\beta }\right].\hfill \end{array}$$

To achieve our purpose, we now put the values $cd=dc=\frac{i-2}{5},\alpha -\beta =\sqrt{5}$ in the above equality. Thus, the proof is completed.  □

Theorem 6. 

(Catalan’s Identity) For $n\ge 1$, the Catalan identity for the Pauli Gaussian Fibonacci quaternions is

$$\begin{array}{c}{Q}_{p}G{F}_n^2-Q_{p}G{F}_{n+r}\times Q_{p}G{F}_{n-r}\hfill \\ ={\left(-1\right)}^{n+1}\frac{\left(2-i\right)}{5}\left[\left(1-{\left(\frac{-3+\sqrt{5}}{2}\right)}^r\right)\widehat{\beta }\widehat{\alpha }+\left(1-{\left(\frac{-3-\sqrt{5}}{2}\right)}^r\right)\widehat{\alpha }\widehat{\beta }\right].\hfill \end{array}$$

Proof. 

Using the Binet formula for Pauli Gaussian Fibonacci quaternions, we have

$$\begin{array}{c}{Q}_{p}G{F}_n^2-Q_{p}G{F}_{n+r}\times Q_{p}G{F}_{n-r}\hfill \\ =\left(c{\alpha }^n\widehat{\alpha }+d{\beta }^n\widehat{\beta }\right)\left(c{\alpha }^n\widehat{\alpha }+d{\beta }^n\widehat{\beta }\right)-\left(c{\alpha }^{n+r}\widehat{\alpha }+d{\beta }^{n+r}\widehat{\beta }\right)\left(c{\alpha }^{n-r}\widehat{\alpha }+d{\beta }^{n-r}\widehat{\beta }\right)\hfill \\ =dc\left({\left(\beta \alpha \right)}^n-{\beta }^{n+r}{\alpha }^{n-r}\right)\widehat{\beta }\widehat{\alpha }+cd\left({\left(\alpha \beta \right)}^n-{\alpha }^{n+r}{\beta }^{n-r}\right)\widehat{\alpha }\widehat{\beta }.\hfill \end{array}$$

Note that we have the following identities

$$\begin{array}{c}\widehat{\alpha }=1+\left(\frac{1+\sqrt{5}}{2}\right){\sigma }_1+\left(\frac{3+\sqrt{5}}{2}\right){\sigma }_2+\left(2+\sqrt{5}\right){\sigma }_3,\hfill \\ \widehat{\beta }=1+\left(\frac{1-\sqrt{5}}{2}\right){\sigma }_1+\left(\frac{3-\sqrt{5}}{2}\right){\sigma }_2+\left(2-\sqrt{5}\right){\sigma }_3,\hfill \\ \widehat{\alpha }\widehat{\beta }=\left(1-\sqrt{5}i\right){\sigma }_1+\left(3-\sqrt{5}i\right){\sigma }_2+\left(4+\sqrt{5}i\right){\sigma }_3,\hfill \\ \widehat{\beta }\widehat{\alpha }=\left(1+\sqrt{5}i\right){\sigma }_1+\left(3+\sqrt{5}i\right){\sigma }_2+\left(4-\sqrt{5}i\right){\sigma }_3.\hfill \end{array}$$

□

It is well-known that if $r=1$ the Cassini identity corresponds to the Cassini identity. Thus, the following corollary can be given.

Corollary 1. 

For $n\ge 1$, the Cassini identity for the Pauli Gaussian Fibonacci quaternions is

$$Q_{p}G{F}_n^2-Q_{p}G{F}_{n+1}\times Q_{p}G{F}_{n-1}={\left(-1\right)}^{n+1}\frac{\left(2-i\right)}{5}\left[\left(\frac{5-\sqrt{5}}{2}\right)\widehat{\beta }\widehat{\alpha }+\left(\frac{5+\sqrt{5}}{2}\right)\widehat{\alpha }\widehat{\beta }\right].$$

Example 2. 

Let $Q_{p}G{F}_5,Q_{p}G{F}_3$ and $Q_{p}G{F}_1$ be Pauli Gaussian Fibonacci quaternions. If we consider Theorem 6 for $n=3$ and $r=2$, the calculations give the following equality

$$\begin{array}{c}{Q}_{p}G{F}_3^2-Q_{p}G{F}_5\times Q_{p}G{F}_1\hfill \\ ={\left(-1\right)}^4\frac{\left(2-i\right)}{5}\left[\left(\frac{-5+3\sqrt{5}}{2}\right)\left(\left(1+\sqrt{5}i\right){\sigma }_1+\left(3+\sqrt{5}i\right){\sigma }_2+\left(4-\sqrt{5}i\right){\sigma }_3\right)\right.\hfill \\ \left.+\left(\frac{-5-3\sqrt{5}}{2}\right)\left(\left(1-\sqrt{5}i\right){\sigma }_1+\left(3-\sqrt{5}i\right){\sigma }_2+\left(4+\sqrt{5}i\right){\sigma }_3\right)\right]\hfill \\ =\frac{\left(2-i\right)}{5}\left[\left(-5+15i\right){\sigma }_1+\left(-15+15i\right){\sigma }_2+\left(-20-15i\right){\sigma }_3\right]\hfill \\ =\left(2-i\right)\left[\left(-1+3i\right){\sigma }_1+\left(-3+3i\right){\sigma }_2+\left(-4-3i\right){\sigma }_3\right].\hfill \end{array}$$

Catalan identity for $n=3$ and $r=2$ obtained from Aydın (see [38]), we found

$$\begin{array}{c}{Q}_p{F}_3^2-Q_p{F}_5\times Q_p{F}_1={\left(-1\right)}^1{F}_2\left[\left(1-i\right){\sigma }_1+\left(3-i\right){\sigma }_2+\left(4+i\right){\sigma }_3\right]\hfill \\ Q_p{F}_3^2-Q_p{F}_5\times Q_p{F}_1=-\left[\left(1-i\right){\sigma }_1+\left(3-i\right){\sigma }_2+\left(4+i\right){\sigma }_3\right].\hfill \end{array}$$

where $Q_p{F}_1,Q_p{F}_3$ and $Q_p{F}_5$ are Pauli Fibonacci quaternions.

Example 3. 

Let $Q_{p}G{F}_3,Q_{p}G{F}_2$ and $Q_{p}G{F}_1$ be Pauli Gaussian Fibonacci quaternions. If we consider Corollary for $n=3$ and $r=1$, the calculations give the following equality

$$\begin{array}{c}{Q}_{p}G{F}_3^2-Q_{p}G{F}_4\times Q_{p}G{F}_2\hfill \\ ={\left(-1\right)}^4\frac{\left(2-i\right)}{5}\left[\left(\frac{5-\sqrt{5}}{2}\right)\left(\left(1+\sqrt{5}i\right){\sigma }_1+\left(3+\sqrt{5}i\right){\sigma }_2+\left(4-\sqrt{5}i\right){\sigma }_3\right)\right.\hfill \\ \left.+\left(\frac{5+\sqrt{5}}{2}\right)\left(\left(1-\sqrt{5}i\right){\sigma }_1+\left(3-\sqrt{5}i\right){\sigma }_2+\left(4+\sqrt{5}i\right){\sigma }_3\right)\right]\hfill \\ =\frac{\left(2-i\right)}{5}\left[\left(5-5i\right){\sigma }_1+\left(15-5i\right){\sigma }_2+\left(20+5i\right){\sigma }_3\right]\hfill \\ =\left(2-i\right)\left[\left(1-i\right){\sigma }_1+\left(3-i\right){\sigma }_2+\left(4+i\right){\sigma }_3\right].\hfill \end{array}$$

On the other hand, if we consider the Cassini identity for $n=3$ and $r=1$ obtained from Aydın (see [38]), we found

$$\begin{array}{c}{Q}_p{F}_3^2-Q_p{F}_4\times Q_p{F}_2={\left(-1\right)}^2{F}_1\left[\left(1-i\right){\sigma }_1+\left(3-i\right){\sigma }_2+\left(4+i\right){\sigma }_3\right]\hfill \\ Q_p{F}_3^2-Q_p{F}_4\times Q_p{F}_2=\left(1-i\right){\sigma }_1+\left(3-i\right){\sigma }_2+\left(4+i\right){\sigma }_3.\hfill \end{array}$$

where $Q_p{F}_1,Q_p{F}_2$ and $Q_p{F}_3$ are Pauli Fibonacci quaternions.

3. Conclusions

In this research, we have extended the quaternions in [38] to the complex case by taking the components of Gaussian Fibonacci and Gaussian Lucas numbers. We have obtained some identities and formulas for these special quaternions, which are specific to Fibonacci quaternions. On the other hand, Pauli matrices and Pauli quaternions have applications in many areas, including quantum mechanics and quantum field theory. We believe that it will be a resource for researchers working in these fields.