Gaussian Chebyshev Polynomials and Their Properties

Abstract

In this study, we define a new family of Gaussian polynomials, called Gaussian Chebyshev polynomials, by extending classical Chebyshev polynomials into the complex domain. These polynomials are characterized by second-order linear recurrence relations, and their connections with the Chebyshev polynomials are established. We also examine properties such as Binet-type formulas and generating functions. Moreover, we characterize some relationships between Gaussian and classical Chebyshev polynomials for the first and second kinds. We obtain some well-known theorems, such as Cassini, Catalan, and d’Ocagne’s theorems, for the first and second kinds. Furthermore, we present important connections among four types of these new polynomials. In the proofs of our results, we utilize the symmetric and antisymmetric properties of the Chebyshev polynomials. Finally, it is shown that Gaussian Chebyshev polynomials are closely related to well-known special sequences such as the Fibonacci, Lucas, Gaussian Fibonacci, and Gaussian Lucas numbers for some specific values of variables.

1. Introduction

Different Gaussian numbers, defined as the extension of well-known number sequences to the complex plane, have been a matter of curiosity for many researchers since they were first investigated in 1832 by Gauss. Horadam [1,2] examined Fibonacci numbers on the complex plane, which were subsequently called Gaussian Fibonacci numbers. He [2] also studied complex Fibonacci polynomials. In 1965, the Gaussian Fibonacci and Gaussian Lucas numbers were studied by the authors of [3]. In 2013, Aşci and Gürel defined the Gaussian Jacobsthal and Gaussian Jacobsthal–Lucas numbers in their study [4]. Then, Halici and Oz [5] introduced the concepts of Gaussian Pell and Gaussian Pell–Lucas numbers. In [6], Özkan and Taştan introduced the notions of Gaussian Fibonacci and Gaussian Lucas polynomials and investigated their fundamental properties, such as the Cassini-type identity, in the Gaussian setting. Then, in another study [7], the Gaussian Jacobsthal and Gaussian Jacobsthal–Lucas polynomials were defined. Moreover, generating functions, Binet’s formulas, explicit formulas, and some properties of these polynomials were investigated. In 2018, the Gaussian Pell polynomials were introduced by Halici and Öz [8], and there has been continued interest and ongoing research on Gaussian numbers and Gaussian polynomials [6,7,8,9,10,11,12,13].

A Gaussian number is a complex number $z=a+ib$, where a and b are integers. Horadam [1,2] and Jordan [3] introduced Fibonacci and Lucas numbers on the complex plane, and they are called Gaussian Fibonacci and Gaussian Lucas numbers:

$$\begin{array}{ccc}\hfill G{F}_n& =& G{F}_{n-1}+G{F}_{n-2};G{F}_0=i,G{F}_1=1;\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill G{L}_n& =& G{L}_{n-1}+G{L}_{n-2};G{L}_0=2-i,G{L}_1=1+2i.\hfill \end{array}$$

(2)

It is clear that

$$\begin{array}{c}\hfill G{F}_n=F_n+i{F}_{n-1},\end{array}$$

(3)

where $F_n$ denotes the Fibonacci sequence defined by the recurrence $F_n=F_{n-1}+F_{n-2}$ with initial conditions $F_0=0$ and $F_1=1$. Moreover,

$$\begin{array}{c}\hfill G{L}_n=L_n+i{L}_{n-1},\end{array}$$

(4)

where $L_n$ denotes the Lucas sequence defined by the recurrence $L_n=L_{n-1}+L_{n-2}$ with initial conditions $L_0=2$ and $L_1=1$.

The Fibonacci polynomials studied by Catalan in 1883 and Lucas polynomials studied by Bicknell in 1970 are defined recursively for $n\ge 2$ by the following relation:

$$\begin{array}{ccc}\hfill f_n\left(x\right)& =& x{f}_{n-1}\left(x\right)+f_{n-2}\left(x\right);f_0\left(x\right)=0,f_1\left(x\right)=1;\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill l_n\left(x\right)& =& x{l}_{n-1}\left(x\right)+l_{n-2}\left(x\right);l_0\left(x\right)=2,l_1\left(x\right)=x.\hfill \end{array}$$

(6)

It is obvious that $f_n\left(1\right)=F_n$ and $l_n\left(1\right)=L_n$, where $f_n$ and $l_n$ are the Fibonacci and Lucas polynomials, and $F_n$ and $L_n$ are the Fibonacci and Lucas numbers, respectively.

Özkan and Taştan [6] introduced the notions of Gaussian Fibonacci and Gaussian Lucas polynomials, establishing connections between these polynomials and the classical Fibonacci and Lucas polynomials.

Gaussian Fibonacci and Gaussian Lucas polynomials are defined for $n\ge 2$ by the following relation (cf., [6]):

$$\begin{array}{ccc}\hfill G{F}_n\left(x\right)& =& xG{F}_{n-1}\left(x\right)+G{F}_{n-2}\left(x\right);G{F}_0\left(x\right)=i,G{F}_1\left(x\right)=1;\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill G{L}_n\left(x\right)& =& xG{L}_{n-1}\left(x\right)+G{L}_{n-2}\left(x\right);G{L}_0\left(x\right)=2-ix,G{L}_1\left(x\right)=x+2i.\hfill \end{array}$$

(8)

Given that generating functions are a fundamental method for solving linear recurrence relations, many studies have focused on deriving generating functions corresponding to various families of Gaussian polynomials.

The ordinary generating functions of Gaussian Fibonacci and Gaussian Lucas polynomials are as follows [13]:

$$\begin{array}{ccc}\hfill \sum _{0\le n<\infty }G{F}_n\left(x\right)z^n& =& {\displaystyle \frac{i-(ix-1)z}{1-xz-z^2}},\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \sum _{0\le n<\infty }G{L}_n\left(x\right)z^n& =& {\displaystyle \frac{2-ix+(i(x^2+2)-x)z}{1-xz-z^2}}.\hfill \end{array}$$

(10)

We encounter Chebyshev polynomials in many branches of mathematics, from approximation theory to number theory. Moreover, they attract substantial attention, as they are closely related to Fibonacci and Lucas sequences [14,15,16,17,18].

In recent years, generalizations that combine classical structures such as Gaussian integers and different polynomials have played an increasingly important role in mathematics. Chebyshev polynomials have a wide range of applications in areas such as approximation theory, numerical analysis, geometry, differential equations, combinatorics, number theory, and cryptography. On the other hand, Gaussian integers, due to their algebraic structure, have fundamental applications in number theory and cryptography. Moreover, for specific complex and real values of a variable, Chebyshev polynomials coincide with well-known integer sequences, such as Fibonacci and Lucas numbers, revealing deep interconnections between orthogonal polynomials and classical number theory.

Therefore, motivated by these developments, a new family of polynomials, termed Gaussian Chebyshev polynomials, is introduced here for the first time. We are interested in investigating Gaussian sequences and Chebyshev polynomials, two topics that play important roles in pure and applied mathematics. Initially, we present and investigate these polynomials at a fundamental level, focusing on their key algebraic properties. This fundamental exploration lays the groundwork for further theoretical development. Given the central role that classical Chebyshev polynomials play across various fields in both pure and applied mathematics, it is anticipated that their Gaussian counterparts will offer similar contributions in a broad range of contexts.

This study provides a comprehensive treatment of recurrence relations, Binet formulas, and generating functions while also establishing notable algebraic identities (Cassini, Catalan, and d’Ocagne) and uncovering significant interrelations with Fibonacci and Lucas polynomials.

There are many studies in the literature on Gaussian generalizations of the Fibonacci and related number sequences and polynomials. Additionally, modern applications of such generalizations in cryptography have also been investigated [19,20]. Therefore, the newly defined Gaussian Chebyshev polynomials exhibit relationships with respect to classical Chebyshev polynomials, Fibonacci-type sequences/polynomials, and their Gaussian analogues, as indicated in Section 5.

In summary, the combination of these two structures enables the definition of new families of polynomials, providing close relations with Fibonacci and Lucas sequences/polynomials.

The breakdown of this manuscript is as follows. In Section 2, we introduce Gaussian Chebyshev polynomials of the first kind and present some results with respect to these polynomials. In Section 3, Gaussian Chebyshev polynomials of the second kind are defined, and we provide the theorems of these polynomials. Moreover, we obtain striking results when both Gaussian Chebyshev polynomials and classical Chebyshev polynomials are investigated together. Furthermore, the relationships between the two new concepts are given. These results strengthen the connections of two types of polynomials. In Section 4, we define third and fourth types of these polynomials. Finally, in Section 5, we determine the relations between Gaussian Chebyshev polynomials and Fibonacci, Lucas, Gaussian Fibonacci, and Gaussian Lucas numbers.

Chebyshev polynomials may be either symmetric or non-symmetric. Specifically, the first and second kinds are symmetric, while the third and fourth kinds are non-symmetric. Among these, the first kind is particularly important due to its simplicity, versatility, and widespread use. Therefore, both we and many other researchers primarily focus on the first and second kinds in our studies.

The Chebyshev polynomials are a family of classical orthogonal polynomials that can be defined both recursively and via trigonometric expressions. Using the substitution $x=cos\theta$, the Chebyshev polynomials of the first $\left(T_n\left(x\right)\right)$, second $\left(U_n\left(x\right)\right)$, third $\left(V_n\left(x\right)\right)$, and fourth $\left(W_n\left(x\right)\right)$ kinds admit the following trigonometric definitions:

$$\begin{array}{ccc}\hfill T_n\left(x\right)& =& cos\left(n\theta \right),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill U_n\left(x\right)& =& {\displaystyle \frac{sin\left(\right(n+1\left)\theta \right)}{sin\theta }},\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill V_n\left(x\right)& =& {\displaystyle \frac{cos\left(\left(n+\frac{1}{2}\right)\theta \right)}{cos\left(\frac{\theta }{2}\right)}},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill W_n\left(x\right)& =& {\displaystyle \frac{sin\left(\left(n+\frac{1}{2}\right)\theta \right)}{sin\left(\frac{\theta }{2}\right)}}.\hfill \end{array}$$

(14)

The four kinds of Chebyshev polynomials are as follows:

$$\begin{array}{ccccccc}\hfill T_0\left(x\right)& =1,\hfill & \hfill T_1\left(x\right)& =x,\hfill & \hfill T_2\left(x\right)& =2x^2-1,\hfill & \hfill \mathrm{etc}.\end{array}$$

(15)

$$\begin{array}{ccccccc}\hfill U_0\left(x\right)& =1,\hfill & \hfill U_1\left(x\right)& =2x,\hfill & \hfill U_2\left(x\right)& =4x^2-1,\hfill & \hfill \mathrm{etc}.\end{array}$$

(16)

$$\begin{array}{ccccccc}\hfill V_0\left(x\right)& =1,\hfill & \hfill V_1\left(x\right)& =2x-1,\hfill & \hfill V_2\left(x\right)& =4x^2-4x+1,\hfill & \hfill \mathrm{etc}.\end{array}$$

(17)

$$\begin{array}{ccccccc}\hfill W_0\left(x\right)& =1,\hfill & \hfill W_1\left(x\right)& =2x+1,\hfill & \hfill W_2\left(x\right)& =4x^2+4x+1,\hfill & \hfill \mathrm{etc}.\end{array}$$

(18)

The Chebyshev polynomials are also defined as the following recurrence relations for $n\ge 1$:

$$\begin{array}{ccc}\hfill T_{n+1}\left(x\right)& =& 2x{T}_n\left(x\right)-T_{n-1}\left(x\right),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill U_{n+1}\left(x\right)& =& 2x{U}_n\left(x\right)-U_{n-1}\left(x\right),\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill V_{n+1}\left(x\right)& =& 2x{V}_n\left(x\right)-V_{n-1}\left(x\right),\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill W_{n+1}\left(x\right)& =& 2x{W}_n\left(x\right)-W_{n-1}\left(x\right),\hfill \end{array}$$

(22)

with $T_0\left(x\right),U_0\left(x\right),V_0\left(x\right),W_0\left(x\right)=1$ and $T_1\left(x\right)=x,U_1\left(x\right)=2x,V_1\left(x\right)=2x-1,W_1\left(x\right)=2x+1$.

The initial conditions of Chebyshev polynomials are directly derived from trigonometric definitions, and they are fixed and made consistent to ensure that the recursive relations work correctly.

Moreover, the general series representations of the first and second kinds are as follows:

$$\begin{array}{ccc}\hfill T_n\left(x\right)& =& \sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k·{\displaystyle \frac{n}{n-k}}\left({\displaystyle \genfrac{}{}{0pt}{}{n-k}{k}}\right){\left(2x\right)}^{n-2k},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill U_n\left(x\right)& =& \sum _{k=0}^{\lfloor n/2\rfloor }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{n-k}{k}}\right){\left(2x\right)}^{n-2k}.\hfill \end{array}$$

(24)

Their generating functions are given as follows:

$$\sum _{0\le n<\infty }{T}_n\left(x\right)t^n={\displaystyle \frac{1-xt}{1-2xt+t^2}},$$

(25)

$$\sum _{0\le n<\infty }{U}_n\left(x\right)t^n={\displaystyle \frac{1}{1-2xt+t^2}},$$

(26)

$$\sum _{0\le n<\infty }{V}_n\left(x\right)t^n={\displaystyle \frac{1-t}{1-2xt+t^2}},$$

(27)

$$\sum _{0\le n<\infty }{W}_n\left(x\right)t^n={\displaystyle \frac{1+t}{1-2xt+t^2}}.$$

(28)

For brevity, we frequently denote Chebyshev polynomials $T_n\left(x\right)$, $U_n\left(x\right)$, $V_n\left(x\right)$, and $W_n\left(x\right)$ as $T_n,U_n,V_n$, and $W_n$, respectively. Where necessary for clarity, we use the full notation, such as $T_n\left(x\right)$.

The Binet-like formulas for these sequences are derived as follows:

$$\begin{array}{ccc}\hfill T_n& =& {\displaystyle \frac{{\alpha }^n+{\beta }^n}{2}},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill U_n& =& {\displaystyle \frac{{\alpha }^{n+1}-{\beta }^{n+1}}{\alpha -\beta }},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill V_n& =& {\displaystyle \frac{{\alpha }^{n+1}-{\beta }^{n+1}}{\alpha -\beta }}-{\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }}=U_n-U_{n-1},\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill W_n& =& {\displaystyle \frac{{\alpha }^{n+1}-{\beta }^{n+1}}{\alpha -\beta }}+{\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }}=U_n+U_{n-1},\hfill \end{array}$$

(32)

where $\alpha ,\beta =x\pm \sqrt{x^2-1}.$

For various properties of these polynomials, we refer to the research of the authors of [21,22,23,24].

Moreover, Chebyshev polynomials are closely related to well-known Fibonacci and Lucas polynomials. The Fibonacci and Lucas sequences and their polynomials are special sequences/polynomials that have attracted substantial interest from many researchers ([25,26,27,28,29,30]), among others.

Now we provide an auxiliary lemma from the authors of [21,22]; the proof for our results is provided in later sections.

Lemma 1

([21,22], Section 19, pp. 371–394). Let $T_n$ and $U_n$ be the nth Chebyshev polynomials of the first and second kinds, respectively. Then, we obtain the following:

(i) 

$T_{m+n}+T_{m-n}=2T_m{T}_n$,

(ii) 

$T_{m+n}-T_{m-n}=2(x^2-1)U_{m-1}{U}_{n-1}$,

(iii) 

$U_{m+n}+U_{m-n}=2U_m{T}_n$,

(iv) 

$U_{m+n}-U_{m-n}=2T_{m+1}{U}_{n-1}$,

(v) 

$T_{m+n}=T_m{U}_n-T_{m-1}{U}_{n-1}$,

(vi) 

$U_{m+n}=U_m{U}_n-U_{m-1}{U}_{n-1}$,

(vii) 

$T_{m+1}{T}_n-T_m{T}_{n+1}=(x^2-1)U_{m-n-1}$,

(viii) 

$U_{m+1}{U}_n-U_m{U}_{n+1}=U_{n-m-1}$,

(ix) 

$T_n^2-(x^2-1)U_{n-1}^2=1$.

(x) 

$T_n^2+T_m^2=1+T_{n+m}{T}_{n-m}$.

2. Gaussian Chebyshev Polynomials of the First Kind

In this section, we define the Gaussian Chebyshev polynomials of the first kind $G{T}_n$ and present some of their basic properties.

Definition 1.

The Gaussian Chebyshev polynomials of the first kind $G{T}_n$ are defined for $n\ge 1$ recurrence by

$$G{T}_{n+1}=2xG{T}_n-G{T}_{n-1},$$

(33)

with initial conditions $G{T}_0=1+xi$ and $G{T}_1=x+i$.

The Gaussian Chebyshev polynomials of the first kind are as follows:

$$\begin{array}{c}\hfill G{T}_0=1+xi,G{T}_1=x+i,G{T}_2=2x^2-1+xi,G{T}_3=4x^3-3x+(2x^2-1)i,\mathrm{etc}.\end{array}$$

(34)

A relation between the Gaussian and classical Chebyshev polynomial of the first kind is given by the following theorem.

Theorem 1.

For $n>1$, we have

$$G{T}_n=T_n+i{T}_{n-1},$$

(35)

where $T_n$ is the nth Chebyshev polynomial of the first kind.

Proof. 

The proof can be observed by induction on n for $n>1$. Take $n=2$; then, from (33), we have

$$\begin{array}{ccc}\hfill G{T}_2& =& 2xG{T}_1-G{T}_0\hfill \\ & =& 2x^2-1+xi\hfill \\ & =& T_2+i{T}_1.\hfill \end{array}$$

Now, suppose that this is true up to the kth natural number. We will show that this is true for $k+1$. By using Equations (33) and (35), we obtain

$$\begin{array}{ccc}\hfill G{T}_{k+1}& =& 2xG{T}_k-G{T}_{k-1}\hfill \\ & =& 2x(T_k+i{T}_{k-1})-(T_{k-1}+i{T}_{k-2})\hfill \\ & =& 2x{T}_k-T_{k-1}+i(2x{T}_{k-1}-T_{k-2})\hfill \\ & =& T_{k+1}+i{T}_k,\hfill \end{array}$$

which completes the proof. □

Moreover, we obtain the Binet-like formula for these polynomials using the following.

Theorem 2.

For $n\ge 0$,

$$G{T}_n={\displaystyle \frac{{\alpha }^n(1+i\beta )}{2}}+{\displaystyle \frac{{\beta }^n(1+i\alpha )}{2}},$$

(36)

where $\alpha ,\beta =x\pm \sqrt{x^2-1}$.

Proof. 

The characteristic equation of the recurrence (33) is $t^2-2xt+1=0$. Then, its solutions are $\alpha ,\beta ={\displaystyle \frac{2x\pm \sqrt{4x^2-4}}{2}}=x\pm \sqrt{x^2-1}$. Thus, the general solution of the recurrence is given by $G{T}_n=C{(x+\sqrt{x^2-1})}^n+D{(x-\sqrt{x^2-1})}^n$. Using the two initial conditions, we obtain $C={\displaystyle \frac{1+i\beta }{2}}$ and $D={\displaystyle \frac{1+i\alpha }{2}}$. Thus, for $n\ge 1$, the desired explicit formula is

$$G{T}_n={\displaystyle \frac{{\alpha }^n(1+i\beta )}{2}}+{\displaystyle \frac{{\beta }^n(1+i\alpha )}{2}}.$$

(37)

□

We now provide the generating function for $G{T}_n$ via the following.

Theorem 3.

The generating function for the sequence ${\left\{G{T}_n\left(x\right)\right\}}_{n=0}^{\infty }$ denoted by $g(y,x)$ is

$$g(y,x)=\sum _{0\le n<\infty }G{T}_n\left(x\right)y^n={\displaystyle \frac{1-xy+(x+y-2x^{2}y)i}{1-2xy+y^2}}.$$

(38)

Proof. 

The generating function for the sequence ${\left\{G{T}_n\left(x\right)\right\}}_{n=0}^{\infty }$ can be written in power series. Then, we have

$$\begin{array}{c}\hfill g(y,x)=\sum _{0\le n<\infty }G{T}_n\left(x\right)y^n=G{T}_0\left(x\right)+G{T}_1\left(x\right)y+G{T}_2\left(x\right)y^2+\dots \end{array}$$

(39)

Multiplying Equation (39) with $-2xy$ and $y^2$, we obtain

$$\begin{array}{c}\hfill -2xyg(y,x)=-2xG{T}_0\left(x\right)y-2xG{T}_1\left(x\right)y^2-2xG{T}_2\left(x\right)y^3+\dots \end{array}$$

(40)

and

$$\begin{array}{c}\hfill y^{2}g(y,x)=G{T}_0\left(x\right)y^2+G{T}_1\left(x\right)y^3+G{T}_2\left(x\right)y^4+\dots \end{array}$$

(41)

Then, adding Equations (39)–(41), we find that

$$\begin{array}{}\text{(42)}& \hfill (1-2xy+y^2)g(y,x)& =& G{T}_0\left(x\right)+(G{T}_1\left(x\right)-2xG{T}_0\left(x\right))y\hfill \text{(43)}& & & +\sum _{2\le n<\infty }(G{T}_n\left(x\right)-2xG{T}_{n-1}\left(x\right)+G{T}_{n-2}\left(x\right))y^n+\dots \hfill \end{array}$$

From Equation (33), we obtain

$$\begin{array}{c}\hfill (1-2xy+y^2)g(y,x)=G{T}_0\left(x\right)+(G{T}_1\left(x\right)-2xG{T}_0\left(x\right))y,\end{array}$$

(44)

Then, by the initial conditions, we have

$$\begin{array}{}\text{(45)}& \hfill (1-2xy+y^2)g(y,x)& =& 1+xi+xy+yi-2xy-2x^{2}yi\hfill \text{(46)}& & =& 1-xy+(x+y-2x^{2}y)i.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill g(y,x)={\displaystyle \frac{1-xy+(x+y-2x^{2}y)i}{1-2xy+y^2}}.\end{array}$$

(47)

Finally, the proof is completed. □

A well-known tool in the study of recursive sequences is the Fibonacci Q-matrix, which is defined by

$$\begin{array}{ccc}\hfill Q& =& \left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right)\hfill \end{array}$$

(48)

and

$$\begin{array}{ccc}\hfill Q^n& =& \left(\begin{array}{cc}{F}_{n+1}& F_n\\ F_n& F_{n-1}\end{array}\right),\hfill \end{array}$$

(49)

where $F_n$ denotes the n-th Fibonacci number. This classical matrix encodes the recurrence relation $F_{n+1}=F_n+F_{n-1}$ and generates the sequence via matrix exponentiation. By encoding these recurrences in matrix form, we not only gain computational convenience but also obtain an algebraic formulation. Analogously, we define two special matrices, $Q\left(x\right)$ and $P\left(x\right)$, that play the role of the Q-matrix of Fibonacci numbers. They reflect the recursive structure that defines the Gaussian Chebyshev polynomials:

$$\begin{array}{ccc}\hfill Q\left(x\right)& =& \left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)\hfill \end{array}$$

(50)

and

$$\begin{array}{ccc}\hfill P\left(x\right)& =& \left(\begin{array}{cc}2x^2-1+xi& x+i\\ x+i& 1+xi\end{array}\right).\hfill \end{array}$$

(51)

Theorem 4.

For $n\ge 1$, we have

$$\begin{array}{ccc}\hfill Q^n\left(x\right)P\left(x\right)& =& \left(\begin{array}{cc}G{T}_{n+2}& G{T}_{n+1}\\ G{T}_{n+1}& G{T}_n\end{array}\right)\hfill \end{array}$$

(52)

where $G{T}_n$ is the nth Gaussian Chebyshev polynomial of the first kind.

Proof. 

We can prove the theorem via induction on n. For $n=1$, we obtain

$$\begin{array}{ccc}\hfill \left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}2x^2-1+xi& x+i\\ x+i& 1+xi\end{array}\right)& =& \left(\begin{array}{cc}4x^3-3x+(2x^2-1)i& 2x^2-1+xi\\ 2x^2-1+xi& x+i\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}G{T}_3& G{T}_2\\ G{T}_2& G{T}_1\end{array}\right).\hfill \end{array}$$

Now, assume that the theorem holds for $n=k$, namely

$$\begin{array}{ccc}\hfill {\left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)}^k\left(\begin{array}{cc}2x^2-1+xi& x+i\\ x+i& 1+xi\end{array}\right)& =& \left(\begin{array}{cc}G{T}_{k+2}& G{T}_{k+1}\\ G{T}_{k+1}& G{T}_k\end{array}\right).\hfill \end{array}$$

Then, for $n=k+1$ we have

$$\begin{array}{ccc}\hfill {\left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)}^{k+1}\left(\begin{array}{cc}2x^2-1+xi& x+i\\ x+i& 1+xi\end{array}\right)& =& \left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)\left(\begin{array}{cc}G{T}_{k+2}& G{T}_{k+1}\\ G{T}_{k+1}& G{T}_k\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}G{T}_{k+3}& G{T}_{k+2}\\ G{T}_{k+2}& G{T}_{k+1}\end{array}\right),\hfill \end{array}$$

This completes the proof. □

Cassini’s identity is well known in the theory of Fibonacci numbers. This theorem can also be used for Gaussian Chebyshev polynomials. Now, we obtain Cassini’s identity for the Gaussian Chebyshev polynomials of the first kind using the following theorem.

Theorem 5.

(Cassini’s Identity): For $n\ge 1$, Cassini’s identity for the Gaussian Chebyshev polynomials of the first kind $G{T}_n$ is given by

$$G{T}_{n+1}G{T}_{n-1}-G{T}_n^2=2xi(x^2-1).$$

(53)

Proof. 

We can prove the theorem via the matrix method. It is clear that the determinants of matrices $Q^{n-1}\left(x\right)$ and $P\left(x\right)$ are as follows:

$$det\left(Q^{n-1}\left(x\right)\right)=det\left({\left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)}^{n-1}\right)={\left(det\left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)\right)}^{n-1}=1^{n-1}=1$$

(54)

and

$$det\left(P\left(x\right)\right)=det\left(\begin{array}{cc}2x^2-1+xi& x+i\\ x+i& 1+xi\end{array}\right)=2xi(x^2-1).$$

(55)

Then, by the previous theorem,

$$\begin{array}{ccc}\hfill Q^{n-1}\left(x\right)P\left(x\right)& =& \left(\begin{array}{cc}G{T}_{n+1}& G{T}_n\\ G{T}_n& G{T}_{n-1}\end{array}\right).\hfill \end{array}$$

(56)

We obtain the determinants of the matrices

$$\begin{array}{}\text{(57)}& \hfill G{T}_{n+1}G{T}_{n-1}-G{T}_n^2& =& det\left(\begin{array}{cc}G{T}_{n+1}& G{T}_n\\ G{T}_n& G{T}_{n-1}\end{array}\right)\hfill \text{(58)}& & =& det\left(Q^{n-1}\left(x\right)P\left(x\right)\right)\hfill \text{(59)}& & =& det\left(Q^{n-1}\left(x\right)\right)det\left(P\left(x\right)\right)\hfill \text{(60)}& & =& 2xi(x^2-1),\hfill \end{array}$$

as claimed. □

Theorem 6.

(Catalan’s Identity): Let $n\ge m\ge 0$. Then,

$$G{T}_{n+m}G{T}_{n-m}-G{T}_n^2=ix(T_{2m}-1).$$

(61)

Proof. 

This theorem can be proved using the Binet formulas. □

Theorem 7.

(d’Ocagne’s identity): For $m>n\ge 0$, we have

$$G{T}_{m+1}G{T}_n-G{T}_{m}G{T}_{n+1}=2xi(x^2-1)U_{m-n-1}.$$

(62)

Proof. 

By Equation (35),

$$\begin{array}{ccc}& & G{T}_{m+1}G{T}_n-G{T}_{m}G{T}_{n+1}\hfill \\ & =& (T_{m+1}+i{T}_m)(T_n+i{T}_{n-1})-(T_m+i{T}_{m-1})(T_{n+1}+i{T}_n)\hfill \\ & =& T_{m+1}{T}_n+i{T}_{m+1}{T}_{n-1}+i{T}_m{T}_n-T_m{T}_{n-1}-T_m{T}_{n+1}-i{T}_m{T}_n-i{T}_{m-1}{T}_{n+1}+T_{m-1}{T}_n\hfill \\ & =& T_n(T_{m+1}+T_{m-1})-T_m(T_{n+1}+T_{n-1})+i(T_{m+1}{T}_{n-1}-T_{m-1}{T}_{n+1})\hfill \end{array}$$

Using Equation (19), we obtain

$$\begin{array}{ccc}& & 2x{T}_m{T}_n-2x{T}_m{T}_n+i((2x{T}_m-T_{m-1})T_{n-1}-T_{m-1}{T}_{n+1})\hfill \\ & =& i(2x{T}_m{T}_{n-1}-T_{m-1}{T}_{n-1}-T_{m-1}{T}_{n+1})\hfill \\ & =& i(2x{T}_m{T}_{n-1}-T_{m-1}(T_{n+1}+T_{n-1}))\hfill \\ & =& i(2x{T}_m{T}_{n-1}-2x{T}_{m-1}{T}_n),\hfill \end{array}$$

by (vii) in Lemma 1, which equals

$$2xi(x^2-1)U_{m-n-1},$$

(63)

This completes the proof. □

In the following sections, we provide some relations between the Gaussian Chebyshev polynomials of the first kind and classical Chebyshev polynomials.

Corollary 1.

For $m\ge n\ge 0$,

$$G{T}_{m+n}+G{T}_{m-n}=2T_{n}G{T}_m.$$

(64)

Proof. 

By Equation (35), we obtain

$$\begin{array}{ccc}\hfill G{T}_{m+n}+G{T}_{m-n}& =& T_{m+n}+T_{m-n}+i(T_{m+n-1}+T_{m-n-1})\hfill \end{array}$$

using Lemma 1 (i), which equals

$$\begin{array}{ccc}& & 2T_m{T}_n+2i{T}_{m-1}{T}_n\hfill \\ & =& 2T_n(T_m+i{T}_{m-1})\hfill \end{array}$$

By Equation (35), it equals

$$\begin{array}{c}\hfill 2T_{n}G{T}_m\end{array}$$

as claimed. □

Corollary 2.

For $m,n\ge 1$,

$$G{T}_{m+n}=U_{n}G{T}_m-U_{n-1}G{T}_{m-1}.$$

Proof. 

By Equation (35), we have

$$\begin{array}{ccc}\hfill G{T}_{m+n}& =& T_{m+n}+i{T}_{m+n-1}\hfill \end{array}$$

using (v) in Lemma 1, which equals

$$\begin{array}{ccc}& & T_m{U}_n-T_{m-1}{U}_{n-1}+i{T}_{m-1}{U}_n-i{T}_{m-2}{U}_{n-1}\hfill \\ & =& U_n(T_m+i{T}_{m-1})-U_{n-1}(T_{m-1}+i{T}_{m-2})\hfill \\ & =& U_{n}G{T}_m-U_{n-1}G{T}_{m-1},\hfill \end{array}$$

This completes the proof. □

Corollary 3.

For $n\ge 1$,

$$G{T}_n^2+G{T}_{n+1}^2=2x(x^2-1)U_{2n-1}+4xi{T}_n^2.$$

(65)

Proof. 

By Equation (35), we have

$$\begin{array}{ccc}\hfill G{T}_n^2+G{T}_{n+1}^2& =& {(T_n+i{T}_{n-1})}^2+{(T_{n+1}+i{T}_n)}^2\hfill \\ & =& (T_n^2+T_{n+1}^2)-(T_{n-1}^2+T_n^2)+2i{T}_n(T_{n-1}+T_{n+1}),\hfill \end{array}$$

If we take $m=n$, $n=n+1$, and $m=n-1$ from Lemma 1 (x), it equals

$$\begin{array}{ccc}& & 1+x{T}_{2n+1}-1-x{T}_{2n-1}+4xi{T}_n^2\hfill \\ & =& x(T_{2n+1}-T_{2n-1})+4xi{T}_n^2\hfill \end{array}$$

Using (ii) in Lemma 1, it equals

$$\begin{array}{ccc}& & 2x(x^2-1)U_{2n-1}{U}_0+4xi{T}_n^2,\hfill \end{array}$$

which further equals

$$\begin{array}{c}\hfill 2x(x^2-1)U_{2n-1}+4xi{T}_n^2,\end{array}$$

as claimed. □

Corollary 4.

For $n\ge 1$,

$$G{T}_{n+1}^2-G{T}_{n-1}^2=4x(x^2-1)(T_{2n-1}+i{U}_{2n-2}).$$

(66)

Proof. 

By Corollary 3, we obtain

$$\begin{array}{ccc}\hfill G{T}_n^2+G{T}_{n+1}^2& =& 2x(x^2-1)U_{2n-1}+4xi{T}_n^2,\hfill \end{array}$$

(67)

$$\begin{array}{ccc}\hfill G{T}_n^2+G{T}_{n-1}^2& =& 2x(x^2-1)U_{2n-3}+4xi{T}_{n-1}^2.\hfill \end{array}$$

(68)

If we subtract Equation (68) from Equation (67), we obtain the following:

$$G{T}_{n+1}^2-G{T}_{n-1}^2=2x(x^2-1)(U_{2n-1}-U_{2n-3})+4xi(T_n^2-T_{n-1}^2),$$

(69)

Using (iv), (i), and (ii) in Lemma 1, we obtain

$$\begin{array}{ccc}\hfill G{T}_{n+1}^2-G{T}_{n-1}^2& =& 4x(x^2-1)T_{2n-1}+2xi(T_{2n}-T_{2n-2})\hfill \\ & =& 4x(x^2-1)T_{2n-1}+4xi(x^2-1)U_{2n-2}\hfill \\ & =& 4x(x^2-1)(T_{2n-1}+i{U}_{2n-2}).\hfill \end{array}$$

Hence, the proof is completed. □

3. Gaussian Chebyshev Polynomials of the Second Kind

In this section, we present the Gaussian Chebyshev polynomials of the second kind $G{U}_n$, which have the same recurrence as $G{T}_n$.

Definition 2.

The Gaussian Chebyshev polynomials of the second kind $G{U}_n$ are defined for $n\ge 1$ recurrence by

$$G{U}_{n+1}=2xG{U}_n-G{U}_{n-1},$$

(70)

with initial conditions $G{U}_0=1$ and $G{U}_1=2x+i$.

The Gaussian Chebyshev polynomials of the second kind are follows:

$$\begin{array}{c}\hfill G{U}_0=1,G{U}_1=2x+i,G{U}_2=4x^2-1+2xi,G{U}_3=8x^3-4x+(4x^2-1)i,\mathrm{etc}.\end{array}$$

(71)

It is observed that

$$G{U}_n=U_n+i{U}_{n-1},$$

(72)

where $U_n$ is the classical nth Chebyshev polynomial of the second kind.

We provide the Binet-like formula of the Gaussian Chebyshev polynomials of the second kind as follows.

Theorem 8.

For $n\ge 0$,

$$G{U}_n={\displaystyle \frac{{\alpha }^n(\alpha +i)}{\alpha -\beta }}-{\displaystyle \frac{{\beta }^n(\beta +i)}{\alpha -\beta }},$$

(73)

where $\alpha ,\beta =x\pm \sqrt{x^2-1}$.

Proof. 

The characteristic equation of recurrence relation (70) is $t^2-2xt+1=0$. Then, its solutions are $\alpha ,\beta ={\displaystyle \frac{2x\pm \sqrt{4x^2-4}}{2}}=x\pm \sqrt{x^2-1}$. Thus, the general solution of the recurrence is given by $G{U}_n=C{(x+\sqrt{x^2-1})}^n+D{(x-\sqrt{x^2-1})}^n$. Using the two initial conditions, we obtain $C={\displaystyle \frac{\alpha +i}{\alpha -\beta }}$ and $D=-{\displaystyle \frac{\beta +i}{\alpha -\beta }}$. Thus, for $n\ge 1$, the desired explicit formula is

$$G{U}_n={\displaystyle \frac{{\alpha }^n(\alpha +i)}{\alpha -\beta }}-{\displaystyle \frac{{\beta }^n(\beta +i)}{\alpha -\beta }}.$$

(74)

□

We now provide the generating function for $G{U}_n$ as follows:

Theorem 9.

The generating function for the Gaussian Chebyshev polynomials of the second-kind sequence ${\left\{G{U}_n\right\}}_{n=0}^{\infty }$ denoted by $g(y,x)$ is as follows:

$$g(y,x)=\sum _{0\le n<\infty }G{U}_n{y}^n={\displaystyle \frac{1+yi}{1-2xy+y^2}}.$$

(75)

Proof. 

This theorem can be proved by writing the generating function as a power series. □

Now, we define two special matrices, $Q\left(x\right)$ and $R\left(x\right)$, that play the role of the Q-matrix of Fibonacci numbers:

$$\begin{array}{ccc}\hfill Q\left(x\right)& =& \left(\begin{array}{cc}2x& -1\\ 1& 0\end{array}\right)\hfill \end{array}$$

(76)

and

$$\begin{array}{ccc}\hfill R\left(x\right)& =& \left(\begin{array}{cc}4x^2-1+2xi& 2x+i\\ 2x+i& 1\end{array}\right).\hfill \end{array}$$

(77)

Theorem 10.

For $n\ge 1$, we have

$$\begin{array}{ccc}\hfill Q^n\left(x\right)R\left(x\right)& =& \left(\begin{array}{cc}G{U}_{n+2}& G{U}_{n+1}\\ G{U}_{n+1}& G{U}_n\end{array}\right),\hfill \end{array}$$

(78)

where $G{U}_n$ is the nth Gaussian Chebyshev polynomial of the second kind.

Now, we provide Cassini’s identity for $G{U}_n$ via the following theorem.

Theorem 11.

(Cassini’s identity): For $n\ge 1$, Cassini’s identity for the Gaussian Chebyshev polynomials of the second kind $G{U}_n$ is given by

$$G{U}_{n+1}G{U}_{n-1}-G{U}_n^2=-2xi.$$

(79)

Proof. 

We can prove the theorem using the matrix method. □

Theorem 12.

(Catalan’s identity): Let $n\ge m\ge 1$. Then,

$$G{U}_{n+m}G{U}_{n-m}-G{U}_n^2=-2xi{U}_{2m-1}.$$

(80)

Proof. 

This theorem can be proven using the Binet formulas. □

Theorem 13.

(d’Ocagne’s Identity): For $m>n\ge 0$, we have

$$G{U}_{m+1}G{U}_n-G{U}_{m}G{U}_{n+1}=2xi{U}_{n-m-1}.$$

(81)

Proof. 

By using Equations (72) and (20) and Lemma 1 (viii), the theorem can be proven. □

With the following theorem, we obtain some relationships between the Gaussian Chebyshev polynomials of the second kind and classical Chebyshev polynomials.

Theorem 14.

The following holds:

(i) 

$$\begin{array}{cccc}\hfill G{U}_{m+n}+G{U}_{m-n}& =2T_{n}G{U}_m,\hfill & \hfill & (m\ge n\ge 0)\hfill \end{array}$$

(ii) 

$$\begin{array}{cccc}\hfill G{U}_{m+n}& =U_{n}G{U}_m-U_{n-1}G{U}_{m-1},\hfill & \hfill & (m,n\ge 1)\hfill \end{array}$$

Proof. 

(i)

By Equation (72),

$$\begin{array}{ccc}& & G{U}_{m+n}+G{U}_{m-n}\hfill \\ & =& (U_{m+n}+i{U}_{m+n-1})+(U_{m-n}+i{U}_{m-n-1})\hfill \\ & =& U_{m+n}+U_{m-n}+i(U_{m+n-1}+U_{m-n-1})\hfill \end{array}$$

Using (iii) in Lemma 1 and Equation (72), we obtain

$$\begin{array}{ccc}& & 2U_m{T}_n+2i{U}_{m-1}{T}_n\hfill \\ & =& 2T_n(U_m+i{U}_{m-1})\hfill \\ & =& 2T_{n}G{U}_m.\hfill \end{array}$$

(ii)

Using Equation (72) and (vi) in Lemma 1, we obtain

$$\begin{array}{ccc}\hfill G{U}_{m+n}& =& U_{m+n}+i{U}_{m+n-1}\hfill \\ & =& U_m{U}_n-U_{m-1}{U}_{n-1}+i(U_{m-1}{U}_n-U_{m-2}{U}_{n-1})\hfill \\ & =& U_n(U_m+i{U}_{m-1})-U_{n-1}(U_{m-1}+i{U}_{m-2})\hfill \\ & =& U_{n}G{U}_m-U_{n-1}G{U}_{m-1}.\hfill \end{array}$$

□

Now, we present the relationships between $G{T}_n$ and $G{U}_n$ with the following theorem:

Theorem 15.

The following holds:

(i) 

$G{U}_n-G{U}_{n-2}=2G{T}_n$, ($n\ge 2$);

(ii) 

$G{U}_n-xG{U}_{n-1}=G{T}_n$, ($n>0$);

(iii) 

$G{U}_{m+n}-G{U}_{m-n}=2U_{n-1}G{T}_{m+1}$, ($m>n\ge 1$);

(iv) 

$G{T}_{m+n}-G{T}_{m-n}=2(x^2-1)U_{n-1}G{U}_{m-1}$. ($m\ge n\ge 1$).

Moreover, with the following theorem, we obtain the Pell-like equation for the Gaussian Chebyshev polynomials.

Theorem 16.

For $n\ge 1$, we have

$$G{T}_n^2-(x^2-1)G{U}_{n-1}^2=2xi.$$

(82)

Proof. 

By Equations (35) and (72),

$$\begin{array}{ccc}& & G{T}_n^2-(x^2-1)G{U}_{n-1}^2\hfill \\ & =& {(T_n+i{T}_{n-1})}^2-(x^2-1){(U_{n-1}+i{U}_{n-2})}^2\hfill \\ & =& (T_n^2-(x^2-1)U_{n-1}^2)-(T_{n-1}^2-(x^2-1)U_{n-2}^2)+2i(T_n{T}_{n-1}-(x^2-1)U_{n-1}{U}_{n-2})\hfill \end{array}$$

Using (ix) in Lemma 1, we obtain

$$\begin{array}{c}\hfill i(2T_n{T}_{n-1}-2(x^2-1)U_{n-1}{U}_{n-2})\end{array}$$

Moreover, using (i) and (ii) in Lemma 1, we obtain

$$\begin{array}{c}\hfill i(T_{2n-1}+T_1-T_{2n-1}+T_1)\end{array}$$

Since $T_1=x$, the proof is completed. □

4. Gaussian Chebyshev Polynomials of the Third and Fourth Kinds

Now, we present the Gaussian Chebyshev polynomials of the third and fourth kinds; they satisfy the same recurrence as $G{T}_n$ and $G{U}_n$. Moreover, we provide generating functions, Binet formulas, and connections with $G{T}_n$ and $G{U}_n$ for the third and fourth kinds without proofs.

Definition 3.

The Gaussian Chebyshev polynomials of the third kind $G{V}_n$ are defined for $n\ge 1$ recurrence by

$$G{V}_{n+1}=2xG{V}_n-G{V}_{n-1},$$

(83)

with initial conditions $G{V}_0=1+i$ and $G{V}_1=2x-1+i$.

The Gaussian Chebyshev polynomials of the third kind are as follows:

$$\begin{array}{c}\hfill G{V}_0=1+i,G{V}_1=2x-1+i,G{V}_2=4x^2-2x-1+(2x-1)i,\mathrm{etc}.\end{array}$$

(84)

It is observed that

$$G{V}_n=V_n+i{V}_{n-1}.$$

(85)

Theorem 17.

The generating function of the sequence ${\left\{G{V}_n\right\}}_{n=0}^{\infty }$ of the Gaussian Chebyshev polynomials of the third kind—denoted by $g(y,x)$—is

$$g(y,x)=\sum _{0\le n<\infty }G{V}_n{y}^n={\displaystyle \frac{1-y+(1+y-2xy)i}{1-2xy+y^2}}.$$

(86)

We obtain the Binet-like formula of the Gaussian Chebyshev polynomials of the third kind via the following.

Theorem 18.

For $n\ge 0$

$$G{V}_n={\displaystyle \frac{{\alpha }^n(\alpha -1+i(1-\beta ))}{\alpha -\beta }}+{\displaystyle \frac{{\beta }^n(1-\beta +i(\alpha -1))}{\alpha -\beta }},$$

(87)

where $\alpha ,\beta =x\pm \sqrt{x^2-1}$.

We provide some relationships of $G{V}_n$ with $G{U}_n$ and $G{T}_n$.

Theorem 19.

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

(i) 

$G{V}_n=G{U}_n-G{U}_{n-1}$;

(ii) 

$G{V}_n+G{V}_{n-1}=2G{T}_n$.

Proof. 

The proof of item (ii) of the theorem is provided. Using $\alpha \beta =1$, where $\alpha \left(x\right),\beta \left(x\right)=x\pm \sqrt{x^2-1}$, the proof is as follows:

$$\begin{array}{ccc}\hfill G{V}_n+G{V}_{n-1}& =& {\alpha }^n{\displaystyle \frac{(\alpha -1+i(1-\beta \left)\right)}{\alpha -\beta }}+{\beta }^n{\displaystyle \frac{(1-\beta +i(\alpha -1\left)\right)}{\alpha -\beta }}\hfill \\ & & +{\alpha }^{n-1}{\displaystyle \frac{(\alpha -1+i(1-\beta \left)\right)}{\alpha -\beta }}+{\beta }^{n-1}{\displaystyle \frac{(1-\beta +i(\alpha -1\left)\right)}{\alpha -\beta }}\hfill \\ & =& {\displaystyle \frac{1}{\alpha -\beta }}[{\alpha }^n(\alpha -\beta )+{\beta }^n(\alpha -\beta )+i{\alpha }^{n-1}\alpha -i{\beta }^{n-1}\beta -i{\alpha }^{n-1}\beta +i{\beta }^{n-1}\alpha ]\hfill \\ & =& {\displaystyle \frac{1}{\alpha -\beta }}[{\alpha }^n(\alpha -\beta )+{\beta }^n(\alpha -\beta )+i{\alpha }^{n-1}(\alpha -\beta )+i{\beta }^{n-1}(\alpha -\beta )]\hfill \\ & =& {\alpha }^n+{\beta }^n+i{\alpha }^{n-1}+i{\beta }^{n-1}\hfill \\ & =& {\alpha }^n(1+i\beta )+{\beta }^n(1+i\alpha )\hfill \\ & =& 2G{T}_n\hfill \end{array}$$

□

Thus, the relationship between $G{V}_n$ and $G{T}_n$ is established using their respective Binet formulas.

Definition 4.

The Gaussian Chebyshev polynomials of the fourth kind $G{W}_n$ are defined for $n\ge 1$ recurrence by

$$G{W}_{n+1}=2xG{W}_n-G{W}_{n-1},$$

(88)

with initial conditions $G{W}_0=1-i$ and $G{W}_1=2x+1+i$.

The Gaussian Chebyshev polynomials of the fourth kind are as follows:

$$\begin{array}{c}\hfill G{W}_0=1-i,G{W}_1=2x+1+i,G{W}_2=4x^2+2x-1+(2x+1)i,\mathrm{etc}.\end{array}$$

(89)

It is observed that

$$G{W}_n=W_n+i{W}_{n-1}.$$

(90)

Theorem 20.

The generating function for the sequence ${\left\{G{W}_n\right\}}_{n=0}^{\infty }$ of Gaussian Chebyshev polynomials of the fourth kind—denoted by $g(y,x)$—is

$$g(y,x)=\sum _{0\le n<\infty }G{W}_n{y}^n={\displaystyle \frac{1+y+(y+2xy-1)i}{1-2xy+y^2}}.$$

(91)

We obtain the Binet-like formula of the Gaussian Chebyshev polynomials of the fourth kind via the following.

Theorem 21.

For $n\ge 0$,

$$G{W}_n={\displaystyle \frac{{\alpha }^n(\alpha +1+i(\beta +1))}{\alpha -\beta }}-{\displaystyle \frac{{\beta }^n(1+\beta +i(\alpha +1))}{\alpha -\beta }},$$

(92)

where $\alpha ,\beta =x\pm \sqrt{x^2-1}$.

Moreover, we have the following relationships of $G{W}_n$ with $G{U}_n$ and $G{T}_n$.

Theorem 22.

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

(i) 

$G{W}_n=G{U}_n+G{U}_{n-1}$;

(ii) 

$G{W}_n-G{W}_{n-1}=2G{T}_n$.

Corollary 5.

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

(i) 

$G{V}_n+G{V}_{n-1}=G{W}_n-G{W}_{n-1}$;

(ii) 

$G{V}_n+G{W}_n=2G{U}_n$.

Proof. 

The proof of item (i) of the Corollary is provided. Using (31), (32), (85), and (90), the following is the proof:

$$\begin{array}{ccc}\hfill G{V}_n+G{V}_{n-1}& =& V_n+i{V}_{n-1}+V_{n-1}+i{V}_{n-2}\hfill \\ & =& U_n-U_{n-1}+i{U}_{n-1}-i{U}_{n-2}+U_{n-1}-U_{n-2}+i{U}_{n-2}-i{U}_{n-3}\hfill \\ & =& (U_n+U_{n-1})-(U_{n-1}+U_{n-2})+i(U_{n-1}+U_{n-2})-i(U_{n-2}+U_{n-3})\hfill \\ & =& W_n-W_{n-1}+i{W}_{n-1}-i{W}_{n-2}\hfill \\ & =& G{W}_n-G{W}_{n-1}.\hfill \end{array}$$

□

5. Relationships Between Gauss–Chebyshev Polynomials and Special Sequences

In this section, it is demonstrated that, under specific conditions, Gaussian Chebyshev polynomials are closely related to the Lucas and Fibonacci polynomials. Furthermore, when the variable is a complex number, such as ${\displaystyle \frac{i}{2}}$, Gaussian Chebyshev polynomials of the first and second kinds reveal connections to the Lucas and Fibonacci numbers. Moreover, we show that these polynomials reduce to sequences such as the Fibonacci and Lucas numbers, in addition to their Gaussian counterparts—Gaussian Fibonacci and Gaussian Lucas numbers—for certain special variable values.

Theorem 23.

Let $G{T}_n\left(x\right)=G{T}_n$ be a Gaussian Chebyshev polynomial of the first kind, and let $l_n\left(x\right)$ be a Lucas polynomial. Then, we obtain

$$G{T}_n\left(x\right)={\displaystyle \frac{i^n}{2}}(l_n(-2ix)+l_{n-1}(-2ix)).$$

(93)

Proof. 

This is clear from Equation (35) and because $l_n\left(2x\right)=2{(-i)}^n{T}_n\left(ix\right)$. □

Theorem 24.

Let $G{U}_n\left(x\right)=G{U}_n$ be a Gaussian Chebyshev polynomial of the second kind, and let $f_n\left(x\right)$ be a Fibonacci polynomial. Then, we obtain

$$G{U}_n\left(x\right)=i^n(f_{n+1}(-2ix)+f_n(-2ix)).$$

(94)

Proof. 

This is clear from Equation (72) and because $f_n\left(2x\right)={(-i)}^{n-1}{U}_{n-1}\left(ix\right)$. □

Furthermore, we have the following results, as $l_n\left(1\right)=L_n$ and $f_n\left(1\right)=F_n$.

Corollary 6.

Let $G{T}_n\left(x\right)=G{T}_n$ be a Gaussian Chebyshev polynomial of the first kind, and let $L_n$ be a Lucas number. Then, we obtain

$$G{T}_n\left({\displaystyle \frac{i}{2}}\right)={\displaystyle \frac{i^n}{2}}{L}_{n+1}.$$

(95)

Corollary 7.

Let $G{U}_n\left(x\right)=G{U}_n$ be a Gaussian Chebyshev polynomial of the second kind, and let $F_n$ be a Fibonacci number. Then, we obtain

$$G{U}_n\left({\displaystyle \frac{i}{2}}\right)=i^n{F}_{n+2}.$$

(96)

The following results present the values of Gaussian Chebyshev polynomials with respect to some special values. We begin with the following lemma.

Lemma 2

([18]). Let $T_n$ be a Chebyshev polynomial of the first kind, and let $F_n$ and $L_n$ be Fibonacci and Lucas numbers, respectively. Then, we obtain the following:

(i) 

$$T_n\left({\displaystyle \frac{3}{2}}\right)={\displaystyle \frac{1}{2}}{L}_{2n}$$

(97)

(ii) 

$$T_n\left({\displaystyle \frac{7}{2}}\right)={\displaystyle \frac{1}{2}}{L}_{4n}$$

(98)

(iii) 

$$T_n\left(\sqrt{5}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{L}_{3n},\hfill & n\mathit{even};\hfill \\ {\displaystyle \frac{\sqrt{5}}{2}}{F}_{3n},\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(99)

(iv) 

$$T_n\left({\displaystyle \frac{\sqrt{5}}{2}}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{L}_n,\hfill & n\mathit{even};\hfill \\ {\displaystyle \frac{\sqrt{5}}{2}}{F}_n,\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(100)

We now present the results.

Corollary 8.

Let $G{T}_n$ be a Gaussian Chebyshev polynomial of the first kind, and let $F_n$, $L_n$, and $G{L}_n$ be the Fibonacci, Lucas, and Gaussian Lucas numbers, respectively. Then, we obtain the following:

(i) 

$$G{T}_n\left({\displaystyle \frac{3}{2}}\right)={\displaystyle \frac{1}{2}}(G{L}_{2n-1}+L_{2n-2})$$

(101)

(ii) 

$$G{T}_n\left({\displaystyle \frac{7}{2}}\right)={\displaystyle \frac{1}{2}}(G{L}_{4n-3}+2L_{4n-2})$$

(102)

(iii) 

$$G{T}_n\left(\sqrt{5}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}(L_{3n}+i\sqrt{5}{F}_{3n-3}),\hfill & n\mathit{even};\hfill \\ {\displaystyle \frac{1}{2}}(\sqrt{5}{F}_{3n}+i{L}_{3n-3}),\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(103)

(iv) 

$$G{T}_n\left({\displaystyle \frac{\sqrt{5}}{2}}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}(L_n+i\sqrt{5}{F}_{n-1}),\hfill & n\mathit{even};\hfill \\ {\displaystyle \frac{1}{2}}(\sqrt{5}{F}_n+i{L}_{n-1}),\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(104)

Proof. 

As a demonstration, Equation (102) is proven. The others can be similarly proven. We use Equation (35), Lemma 2, and Equation (4), and we obtain the following:

$$\begin{array}{ccc}\hfill G{T}_n\left({\displaystyle \frac{7}{2}}\right)& =& T_n\left({\displaystyle \frac{7}{2}}\right)+i{T}_{n-1}\left({\displaystyle \frac{7}{2}}\right)\hfill \\ & =& {\displaystyle \frac{1}{2}}(L_{4n}+i{L}_{4n-4})\hfill \\ & =& {\displaystyle \frac{1}{2}}(L_{4n-1}+L_{4n-2}+i{L}_{4n-4})\hfill \\ & =& {\displaystyle \frac{1}{2}}(L_{4n-2}+L_{4n-3}+L_{4n-2}+i{L}_{4n-4})\hfill \\ & =& {\displaystyle \frac{1}{2}}(G{L}_{4n-3}+2L_{4n-2}).\hfill \end{array}$$

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Lemma 3

([18]). Let $U_n$ be the Chebyshev polynomial of the second kind, and let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, respectively. Then, we obtain the following:

(i) 

$$U_n\left({\displaystyle \frac{3}{2}}\right)=F_{2n+2}$$

(105)

(ii) 

$$U_n\left({\displaystyle \frac{7}{2}}\right)={\displaystyle \frac{1}{3}}{F}_{4n+4}$$

(106)

(iii) 

$$U_n\left(\sqrt{5}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{4}}{L}_{3n+3},\hfill & n\mathit{even};\hfill \\ {\displaystyle \frac{\sqrt{5}}{4}}{F}_{3n+3},\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(107)

(iv) 

$$U_n\left({\displaystyle \frac{\sqrt{5}}{2}}\right)=\left\{\begin{array}{cc}{L}_{n+1},\hfill & n\mathit{even};\hfill \\ \sqrt{5}{F}_{n+1},\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(108)

Corollary 9.

Let $G{U}_n$ be a Gaussian Chebyshev polynomial of the second kind, and let $F_n$, $L_n$, and $G{F}_n$ be the Fibonacci, Lucas, and Gaussian Fibonacci numbers, respectively. Then, we obtain the following:

(i) 

$$G{U}_n\left({\displaystyle \frac{3}{2}}\right)=(G{F}_{2n+1}+F_{2n})$$

(109)

(ii) 

$$G{U}_n\left({\displaystyle \frac{7}{2}}\right)={\displaystyle \frac{1}{3}}(G{F}_{4n+4}+i{F}_{4n})$$

(110)

(iii) 

$$G{U}_n\left(\sqrt{5}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{4}}(L_{3n+3}+i\sqrt{5}{F}_{3n}),\hfill & n\mathit{even};\hfill \\ {\displaystyle \frac{1}{4}}(\sqrt{5}{F}_{3n+3}+i{L}_{3n}),\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(111)

(iv) 

$$G{U}_n\left({\displaystyle \frac{\sqrt{5}}{2}}\right)=\left\{\begin{array}{cc}{L}_{n+1}+i\sqrt{5}{F}_n,\hfill & n\mathit{even};\hfill \\ \sqrt{5}{F}_{n+1}+i{L}_n,\hfill & n\mathit{odd}.\hfill \end{array}\right.$$

(112)

Proof. 

As a demonstration, Equation (109) is proven. The others can be similarly proven. We use Equation (72), Lemma 3, and Equation (3), and we obtain the following:

$$\begin{array}{ccc}\hfill G{U}_n\left({\displaystyle \frac{3}{2}}\right)& =& U_n\left({\displaystyle \frac{3}{2}}\right)+i{U}_{n-1}\left({\displaystyle \frac{3}{2}}\right)\hfill \\ & =& (F_{2n+2}+i{F}_{2n})\hfill \\ & =& (F_{2n+1}+F_{2n}+i{F}_{2n})\hfill \\ & =& (G{F}_{2n+1}+F_{2n})\hfill \end{array}$$

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6. Conclusions

In this study, four new polynomials were introduced using the definition of Chebyshev polynomials and Gaussian sequences. A fundamental investigation of these polynomials was carried out, focusing on their essential algebraic properties. We explained the relationships between Gaussian Chebyshev polynomials and classical Chebyshev polynomials. Many important results for these polynomials were presented, and the relationships between the Gaussian Chebyshev polynomials of the first and second kinds were provided. Identities such as Cassini, Catalan, and d’Ocagne are fundamental for understanding the intrinsic properties of Fibonacci and other related sequences. Thus, we presented Binet-like formulas and Cassini-like, Catalan-like, and d’Ocagne-like identities for these polynomials. To conclude, this study provided a comprehensive treatment of recurrence relations, Binet formulas, and generating functions while also establishing notable algebraic identities (Cassini, Catalan, and d’Ocagne) and uncovering significant interrelations between Fibonacci and Lucas polynomials. There are many studies in the literature on Chebyshev polynomials and Gaussian generalizations of Fibonacci and other related number sequences/polynomials. Additionally, modern applications of such generalizations in cryptography have also been investigated [19,20]. Therefore, the newly defined Gauss–Chebyshev polynomials are related not only to the classical Chebyshev polynomials but also to Fibonacci-type sequences and polynomials, along with their Gaussian analogues, as demonstrated through several examples in Section 5. Classical Chebyshev polynomials have a wide range of applications in areas such as approximation theory, numerical analysis, geometry, differential equations, combinatorics, number theory, and cryptography. Similarly to Chebyshev polynomials, Gaussian Chebyshev polynomials have the potential to be effectively applied in areas such as approximation theory, differential equations, and number theory. This study is expected to encourage further investigation into various properties of polynomials and the relationships between them. The combined study of Gaussian sequences and Chebyshev polynomials is expected to yield valuable insights and advancements in both theoretical and applied mathematics.