On r-Circulant Matrices with Higher-Order Fibonacci Numbers

Abstract

In this paper, we introduce and investigate a new class of r-circulant matrices whose entries are generated by higher-order Fibonacci numbers. Explicit representations of the eigenvalues of these matrices are derived by means of the Binet formula together with the structural properties of r-circulant matrices. Based on these representations, a closed-form expression for the determinant is obtained. In addition, several summation identities involving higher-order Fibonacci numbers are established, including formulas for partial sums, sums of squares, and weighted sums. These identities play a fundamental role in the derivation of the norm expressions and spectral estimates of the matrices. Furthermore, several matrix norms, including the Euclidean (Frobenius) norm, the 1-norm, the ∞-norm, and the spectral norm, are investigated in detail. Lower and upper bounds for the spectral norm are obtained for both cases $\left|r\right|\ge 1$ and $\left|r\right|<1$ by employing Hadamard product techniques and classical norm inequalities. Finally, numerical examples are presented to illustrate and validate the theoretical results.

1. Introduction

Circulant matrices represent a significant category of structured matrices that naturally emerge in various domains of applied and pure mathematics, such as numerical analysis, signal processing, coding theory, probability, and linear system theory. Owing to their highly regular structure, circulant-type matrices facilitate explicit spectral analysis, rendering them particularly appealing for both theoretical exploration and practical application. In particular, these matrices possess an inherent cyclic (shift-invariant) symmetry, whereby each row is obtained from the previous one by a fixed shift, leading to a natural diagonalization in terms of roots of unity. This symmetry plays a fundamental role in determining their spectral properties and enables the derivation of explicit eigenvalue formulas. A thorough examination of circulant matrices and their algebraic properties is available in the monograph by Davis [1], while general matrix-theoretic methodologies are comprehensively documented in the seminal works of Horn and Johnson [2,3].

An $n\times n$ matrix $C_r$ is called an r-circulant matrix if it is of the following form:

$$C_r=\left(\begin{array}{cccccc}{c}_0& c_1& c_2& \dots & c_{n-2}& c_{n-1}\\ r{c}_{n-1}& c_0& c_1& \dots & c_{n-3}& c_{n-2}\\ r{c}_{n-2}& r{c}_{n-1}& c_0& \dots & c_{n-4}& c_{n-3}\\ ⋮& ⋮& ⋮& & ⋮& ⋮\\ r{c}_1& r{c}_2& r{c}_3& \dots & r{c}_{n-1}& c_0\end{array}\right).$$

Since $C_r$ is completely determined by its first row and the parameter r, it is customary to denote:

$$C_r={Circ}_r(c_0,c_1,\dots ,c_{n-1}).$$

In the special case $r=1$, the matrix $C_r$ reduces to the classical circulant matrix:

$$C:=Circ(c_0,c_1,\dots ,c_{n-1}):=\left(\begin{array}{cccccc}{c}_0& c_1& c_2& \dots & c_{n-2}& c_{n-1}\\ c_{n-1}& c_0& c_1& \dots & c_{n-3}& c_{n-2}\\ c_{n-2}& c_{n-1}& c_0& \dots & c_{n-4}& c_{n-3}\\ ⋮& ⋮& ⋮& & ⋮& ⋮\\ c_1& c_2& c_3& \dots & c_{n-1}& c_0\end{array}\right),$$

which has been extensively studied in the literature.

One of the fundamental advantages of r-circulant matrices is that their eigenvalues admit closed-form expressions. Specifically, in [4,5] the eigenvalues of $C_r$ are given by:

$${\lambda }_i\left(C_r\right)={\displaystyle \sum _{j=0}^{n-1}}{c}_j{\left(\rho {\omega }^{-i}\right)}^j,i=0,1,\dots ,n-1,$$

(1)

where $\omega$ is a primitive nth root of unity and $\rho$ is an nth root of r. This explicit diagonalization facilitates the analysis of spectral properties, determinants, inverses, and various matrix norms.

In recent years, considerable attention has been devoted to the study of circulant and r-circulant matrices whose entries are defined by special number sequences. Such matrices provide a natural bridge between number theory and matrix analysis and have been shown to yield elegant closed-form results for eigenvalues, determinants, and norm estimates.

The investigation of matrix norms for circulant matrices with specific numerical sequences as entries has garnered significant scholarly interest over the past two decades. One of the initial comprehensive studies was conducted by Solak [6], who derived norm estimates for circulant matrices generated by classical Fibonacci and Lucas numbers. Subsequently, Ipek [7] calculated exact spectral norms for analogous matrices, while Shen and Cen [8] expanded these findings to r-circulant matrices, providing both upper and lower bounds for their norms. In this context, He et al. [9] conducted an investigation into r-circulant matrices characterized by entries involving Fibonacci and Lucas numbers, as well as their various combinations. They established upper bound estimates for the spectral norm and demonstrated that these results offer more precise bounds than those previously obtained by Solak, Shen, and Cen. Moreover, the efficacy and accuracy of these enhanced bounds were corroborated by numerical examples. Related identities were established in [10,11], where circulant-type matrices incorporating binomial coefficients and harmonic numbers were analyzed. Specifically, the spectral norms of even-order r-circulant matrices were examined in [12].

In parallel with these developments, several scholars have investigated circulant and r-circulant matrices associated with generalized Fibonacci-type sequences. Bahşi and Solak [13] examined the norms of circulant and r-circulant matrices whose elements are hyper-Fibonacci and hyperharmonic Fibonacci numbers. Related research involving generalized Horadam sequences and k-Horadam numbers was conducted by Alptekin et al. [14] and Yazlik and Taskara [15]. More recently, circulant matrices generated by bi-periodic Fibonacci and Lucas numbers [16], as well as matrices involving Mersenne and Fermat numbers [5], have been analyzed from a spectral norm perspective.

Another significant area of research involves the study of circulant matrices defined through geometric or mixed constructions. In this regard, Kızılateş and Tuglu [17,18] have developed norm estimates for r-circulant and geometric circulant matrices associated with hyperharmonic Fibonacci numbers. Further advancements have been made by Shi and Kızılateş [19], who investigated the spectral norms of $RFPrLrR$ circulant matrices. Additionally, recent contributions by Anđelić et al. [20] have introduced r-min and r-max matrices incorporating harmonic higher-order Gauss Fibonacci numbers, underscoring the increasing interest in integrating structured matrices with higher-order Fibonacci-type sequences.

Despite this growing body of work, the spectral and norm properties of r-circulant matrices generated by higher-order Fibonacci numbers remain largely unexplored. These numbers, introduced by Pashaev and Nalci [21], naturally generalize the classical Fibonacci sequence and arise in various contexts of mathematical physics and discrete dynamical systems. Their rich algebraic structure suggests that matrices constructed from higher-order Fibonacci numbers may exhibit new and nontrivial spectral behavior.

Motivated by these observations, the aim of this paper is to introduce and systematically study r-circulant matrices whose entries are given by higher-order Fibonacci numbers. In particular, we derive explicit formulas for their eigenvalues and determinants, establish bounds for the 1-, ∞-, Euclidean (Frobenius), and spectral norms, and analyze how these properties depend on the parameters r, s, and the matrix dimension n. Our results extend and unify several known results for classical Fibonacci-based circulant matrices and provide new insights into the interplay between generalized Fibonacci sequences and structured matrix theory.

The structure of this paper is as follows: Section 2 presents the necessary preliminaries, including the basic definitions and properties of higher-order Fibonacci numbers, along with fundamental concepts from matrix norms and auxiliary results. In Section 3, we introduce the r-circulant matrices generated by higher-order Fibonacci numbers and establish several essential summation identities that underpin our analysis. Section 4 is dedicated to the spectral investigation of these matrices, where explicit formulas for eigenvalues and determinants are derived, and various matrix norms are examined in detail, including precise bounds for the spectral norm under different conditions on the parameter r. In Section 5, numerical examples are provided to illustrate and validate the theoretical findings. Finally, Section 6 concludes this paper with a summary of the main results and potential directions for future research.

2. Preliminaries

In this section, we present the essential definitions and supplementary results that will be referenced throughout this paper.

The Fibonacci sequence ${\left\{F_n\right\}}_{n\ge 0}$ is defined by the recurrence relation:

$$F_n=F_{n-1}+F_{n-2},n\ge 2,$$

(2)

with initial conditions $F_0=0$ and $F_1=1$. Similarly, the Lucas sequence ${\left\{L_n\right\}}_{n\ge 0}$ satisfies:

$$L_n=L_{n-1}+L_{n-2},n\ge 2,$$

(3)

with $L_0=2$ and $L_1=1$.

Let $\alpha$ and $\beta$ be the roots of the characteristic equation

$$x^2-x-1=0,$$

that is,

$$\alpha ={\displaystyle \frac{1+\sqrt{5}}{2}},\beta ={\displaystyle \frac{1-\sqrt{5}}{2}}.$$

Then, the classical Binet formulas for Fibonacci and Lucas numbers are given by

$$F_n={\displaystyle \frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }},$$

(4)

and

$$L_n={\alpha }^n+{\beta }^n,$$

(5)

respectively.

Following Pashaev and Nalci [21], the higher-order Fibonacci numbers ($s\ge 1$) are defined by:

$$F_n^{\left(s\right)}={\displaystyle \frac{F_{ns}}{F_s}}={\displaystyle \frac{{\left({\alpha }^s\right)}^n-{\left({\beta }^s\right)}^n}{{\alpha }^s-{\beta }^s}},n\in \mathbb{N}.$$

(6)

Since $F_{ns}$ is divisible by $F_s$, each $F_n^{\left(s\right)}$ is an integer. Clearly, the classical Fibonacci sequence is recovered for $s=1$.

Higher-order Fibonacci numbers (also called Fibonacci divisors) and their properties have been studied extensively, particularly in connection with applications in mathematical physics; see, for instance, Pashaev [22] and the references therein.

Next, we recall several matrix norms that will be used throughout this paper. Let $F=\left(f_{ij}\right)$ be a complex square matrix of order n. The Euclidean (Frobenius) norm, spectral norm, 1-norm, and ∞-norm of F are defined, respectively, by

$${\parallel F\parallel }_E={\left({\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{\left|f_{ij}\right|}^2\right)}^{1/2},$$

(7)

$${\parallel F\parallel }_2=\sqrt{{\lambda }_{max}\left(F^{\ast }F\right)},$$

(8)

$${\parallel F\parallel }_1=\underset{1\le j\le n}{max}{\displaystyle \sum _{i=1}^n}|f_{ij}{|\mathrm{and}\parallel F\parallel }_{\infty }=\underset{1\le i\le n}{max}{\displaystyle \sum _{j=1}^n}\left|f_{ij}\right|,$$

(9)

where $F^{\ast }$ denotes the conjugate transpose of F and ${\lambda }_{max}(·)$ is the largest eigenvalue.

The Euclidean and spectral norms satisfy the well-known inequalities:

$${\displaystyle \frac{1}{\sqrt{n}}}{\parallel F\parallel }_E\le {\parallel F\parallel }_2\le {\parallel F\parallel }_E.$$

(10)

The Hadamard product of two matrices $F=\left(f_{ij}\right)$ and $Q=\left(q_{ij}\right)$ of the same size is defined by:

$$F\circ Q=\left(f_{ij}{q}_{ij}\right).$$

The following inequalities will be essential in estimating the spectral norms of Hadamard products.

Lemma 1

([3]). Let F and Q be two $m\times n$ matrices. Then:

$${\parallel F\circ Q\parallel }_2\le {\parallel F\parallel }_2{\parallel Q\parallel }_2.$$

Lemma 2

([3]). Let $F=\left(f_{ij}\right)$ and $Q=\left(q_{ij}\right)$ be two $m\times n$ matrices. Then:

$${\parallel F\circ Q\parallel }_2\le r_1\left(F\right)c_1\left(Q\right),$$

where

$$r_1\left(F\right)=\underset{1\le i\le m}{max}{\left({\displaystyle \sum _{j=1}^n}{\left|f_{ij}\right|}^2\right)}^{1/2}\mathrm{and}{c}_1\left(Q\right)=\underset{1\le j\le n}{max}{\left({\displaystyle \sum _{i=1}^m}{\left|q_{ij}\right|}^2\right)}^{1/2}.$$

Finally, we recall a classical product identity that will be used in the derivation of eigenvalues and determinants of r-circulant matrices.

Lemma 3

([5,23]). Let $r\in \mathbb{C}$ be the circulant parameter and let ρ be a fixed nth root of r, that is, ${\rho }^n=r$, and let ω be a primitive nth root of unity. Then, for any $x,y\in \mathbb{C}$:

$${\displaystyle \prod _{j=0}^{n-1}}\left(x-y\rho {\omega }^{-j}\right)=x^n-r{y}^n.$$

Lemma 4.

Let $n,s\in \mathbb{N}$ with $n\ge 2$. Define:

$$\Omega (n,s):={\displaystyle \sum _{l=1}^{n-1}}{(-1)}^{ls}.$$

Then:

$$\Omega (n,s)=\left\{\begin{array}{cc}n-1,\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ -1,\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ 0,\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(11)

Lemma 5.

Let $n,s\in \mathbb{N}$ with $n\ge 2$. Define:

$$\Gamma (n,s):={\displaystyle \sum _{l=1}^{n-1}}{(-1)}^{ls}l.$$

Then:

$$\Gamma (n,s)=\left\{\begin{array}{cc}{\displaystyle \frac{(n-1)n}{2}},\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ -{\displaystyle \frac{n}{2}},\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ {\displaystyle \frac{n-1}{2}},\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(12)

Lemma 6.

Let $\gamma \in \mathbb{C}$ with $\gamma \ne \pm 1$ and let $n\ge 2$. Then:

$${\displaystyle \sum _{l=1}^{n-1}}l{\gamma }^l=\gamma {\displaystyle \frac{d}{d\gamma }}\left({\displaystyle \sum _{l=1}^{n-1}}{\gamma }^l\right)=\gamma ·{\displaystyle \frac{1-n{\gamma }^{n-1}+(n-1){\gamma }^n}{{(1-\gamma )}^2}},$$

(13)

where the right-hand side is obtained by differentiating the geometric sum ${\sum }_{l=1}^{n-1}{\gamma }^l={\displaystyle \frac{\gamma (1-{\gamma }^{n-1})}{1-\gamma }}$ with respect to γ.

3. Definition and Auxiliary Summation Identities

In this section, we define the r-circulant matrix associated with higher-order Fibonacci numbers that will be studied throughout the paper. We also present several useful summation formulas involving higher-order Fibonacci numbers, which will be used extensively in the proofs of the main results. These preliminary materials provide the notation and technical tools required in the subsequent sections.

Definition 1.

Let ${\left\{F_n^{\left(s\right)}\right\}}_{n\ge 0}$ denote the sequence of higher-order Fibonacci numbers of order s, and let $r\in \mathbb{C}$. The r-circulant matrix associated with the higher-order Fibonacci numbers is defined by:

$$F_r:=\left(\begin{array}{cccccc}{F}_0^{\left(s\right)}& F_1^{\left(s\right)}& F_2^{\left(s\right)}& \cdots & F_{n-2}^{\left(s\right)}& F_{n-1}^{\left(s\right)}\\ r{F}_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}& F_1^{\left(s\right)}& \cdots & F_{n-3}^{\left(s\right)}& F_{n-2}^{\left(s\right)}\\ r{F}_{n-2}^{\left(s\right)}& r{F}_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}& \cdots & F_{n-4}^{\left(s\right)}& F_{n-3}^{\left(s\right)}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ r{F}_2^{\left(s\right)}& r{F}_3^{\left(s\right)}& r{F}_4^{\left(s\right)}& \cdots & F_0^{\left(s\right)}& F_1^{\left(s\right)}\\ r{F}_1^{\left(s\right)}& r{F}_2^{\left(s\right)}& r{F}_3^{\left(s\right)}& \cdots & r{F}_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}\end{array}\right).$$

Equivalently, the matrix $F_r$ can be written in compact form as follows:

$$F_r={Circ}_r\left(F_0^{\left(s\right)},F_1^{\left(s\right)},F_2^{\left(s\right)},\dots ,F_{n-1}^{\left(s\right)}\right),$$

where ${Circ}_r(·)$ denotes the r-circulant matrix generated by its first row.

To facilitate the subsequent analysis of eigenvalues and matrix norms, we first present several auxiliary theorems concerning higher-order Fibonacci numbers that will be required in the proofs of the main results.

Theorem 1.

Let $n,s\in \mathbb{N}$ with $n\ge 1$, and let ${\left\{F_n^{\left(s\right)}\right\}}_{n\ge 0}$ denote the sequence of higher-order Fibonacci numbers of order s. Then, the following summation formula holds:

$${\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}={\displaystyle \frac{1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}}{1-L_s+{(-1)}^s}},$$

(14)

where $L_s$ denotes the sth Lucas number.

Proof. 

Using Equation (6), we obtain:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}& ={\displaystyle \sum _{l=0}^{n-1}}{\displaystyle \frac{{\alpha }^{ls}-{\beta }^{ls}}{{\alpha }^s-{\beta }^s}}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\alpha }^s-{\beta }^s}}\left({\displaystyle \sum _{l=0}^{n-1}}{\alpha }^{ls}-{\displaystyle \sum _{l=0}^{n-1}}{\beta }^{ls}\right).\hfill \end{array}$$

Since both sums are geometric series, we have:

$${\displaystyle \sum _{l=0}^{n-1}}{\alpha }^{ls}={\displaystyle \frac{1-{\alpha }^{sn}}{1-{\alpha }^s}}\mathrm{and}{\displaystyle \sum _{l=0}^{n-1}}{\beta }^{ls}={\displaystyle \frac{1-{\beta }^{sn}}{1-{\beta }^s}}.$$

Substituting these expressions yields:

$${\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}={\displaystyle \frac{1}{{\alpha }^s-{\beta }^s}}\left({\displaystyle \frac{1-{\alpha }^{sn}}{1-{\alpha }^s}}-{\displaystyle \frac{1-{\beta }^{sn}}{1-{\beta }^s}}\right).$$

Combining the two fractions over the common denominator $(1-{\alpha }^s)(1-{\beta }^s)$ gives:

$${\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}={\displaystyle \frac{(1-{\alpha }^{sn})(1-{\beta }^s)-(1-{\beta }^{sn})(1-{\alpha }^s)}{({\alpha }^s-{\beta }^s)(1-{\alpha }^s)(1-{\beta }^s)}}.$$

Expanding the numerator:

$$\begin{array}{cc}\hfill & (1-{\alpha }^{sn})(1-{\beta }^s)-(1-{\beta }^{sn})(1-{\alpha }^s)\hfill \\ \hfill & =1-{\beta }^s-{\alpha }^{sn}+{\alpha }^{sn}{\beta }^s-1+{\alpha }^s+{\beta }^{sn}-{\beta }^{sn}{\alpha }^s\hfill \\ \hfill & =({\alpha }^s-{\beta }^s)-({\alpha }^{sn}-{\beta }^{sn})+{\alpha }^s{\beta }^s\left({\alpha }^{s(n-1)}-{\beta }^{s(n-1)}\right).\hfill \end{array}$$

Using ${\alpha }^s{\beta }^s={\left(\alpha \beta \right)}^s={(-1)}^s$ and dividing by $({\alpha }^s-{\beta }^s)$, the numerator becomes:

$$({\alpha }^s-{\beta }^s)\left(1-{\displaystyle \frac{{\alpha }^{sn}-{\beta }^{sn}}{{\alpha }^s-{\beta }^s}}+{(-1)}^s{\displaystyle \frac{{\alpha }^{s(n-1)}-{\beta }^{s(n-1)}}{{\alpha }^s-{\beta }^s}}\right)=({\alpha }^s-{\beta }^s)\left(1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}\right).$$

For the denominator, we use ${\alpha }^s+{\beta }^s=L_s$ and ${\alpha }^s{\beta }^s={(-1)}^s$:

$$(1-{\alpha }^s)(1-{\beta }^s)=1-({\alpha }^s+{\beta }^s)+{\alpha }^s{\beta }^s=1-L_s+{(-1)}^s.$$

Cancelling the common factor $({\alpha }^s-{\beta }^s)$, we obtain:

$${\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}={\displaystyle \frac{1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}}{1-L_s+{(-1)}^s}},$$

which completes the proof. □

Theorem 2.

Let $n,s\in \mathbb{N}$ with $n\ge 2$, and let ${\left\{F_n^{\left(s\right)}\right\}}_{n\ge 0}$ denote the higher-order Fibonacci numbers of order s. Then, the following summation formula holds:

$${\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2=\left\{\begin{array}{cc}{\displaystyle \frac{L_{2sn}-L_{2s(n-1)}+(2n-1)\left(2-L_{2s}\right)}{{\left(L_{2s}-2\right)}^2}},\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ {\displaystyle \frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{\left(L_{2s}-2\right)\left(L_{2s}+2\right)}},\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ {\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{\left(L_{2s}-2\right)\left(L_{2s}+2\right)}},\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(15)

Here, $L_k$ denotes the kth Lucas number.

Proof. 

Using Equation (6), we have:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \sum _{l=1}^{n-1}}{\left({\displaystyle \frac{{\alpha }^{ls}-{\beta }^{ls}}{{\alpha }^s-{\beta }^s}}\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{{({\alpha }^s-{\beta }^s)}^2}}\left({\displaystyle \sum _{l=1}^{n-1}}{\alpha }^{2ls}-2{\displaystyle \sum _{l=1}^{n-1}}{\left(\alpha \beta \right)}^{ls}+{\displaystyle \sum _{l=1}^{n-1}}{\beta }^{2ls}\right).\hfill \end{array}$$

Since $\alpha \beta =-1$, we have ${\left(\alpha \beta \right)}^{ls}={(-1)}^{ls}$. Moreover, the first and third sums are geometric series, while the middle term can be expressed in terms of $\Omega (n,s)$ defined in Lemma 4. Hence:

$${\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2={\displaystyle \frac{1}{{({\alpha }^s-{\beta }^s)}^2}}\left({\displaystyle \sum _{l=1}^{n-1}}{\alpha }^{2ls}+{\displaystyle \sum _{l=1}^{n-1}}{\beta }^{2ls}-2\Omega (n,s)\right).$$

We now evaluate the two geometric sums explicitly. Since ${\alpha }^{2s}\ne 1$ and ${\beta }^{2s}\ne 1$, we have:

$${\displaystyle \sum _{l=1}^{n-1}}{\alpha }^{2ls}={\displaystyle \frac{{\alpha }^{2s}(1-{\alpha }^{2s(n-1)})}{1-{\alpha }^{2s}}}\mathrm{and}{\displaystyle \sum _{l=1}^{n-1}}{\beta }^{2ls}={\displaystyle \frac{{\beta }^{2s}(1-{\beta }^{2s(n-1)})}{1-{\beta }^{2s}}}.$$

Adding these over the common denominator $(1-{\alpha }^{2s})(1-{\beta }^{2s})$ gives:

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{l=1}^{n-1}}{\alpha }^{2ls}+{\displaystyle \sum _{l=1}^{n-1}}{\beta }^{2ls}\hfill \\ \hfill & ={\displaystyle \frac{{\alpha }^{2s}(1-{\alpha }^{2s(n-1)})(1-{\beta }^{2s})+{\beta }^{2s}(1-{\beta }^{2s(n-1)})(1-{\alpha }^{2s})}{(1-{\alpha }^{2s})(1-{\beta }^{2s})}}.\hfill \end{array}$$

Expanding the numerator:

$$\begin{array}{cc}\hfill & {\alpha }^{2s}(1-{\beta }^{2s}-{\alpha }^{2s(n-1)}+{\alpha }^{2s(n-1)}{\beta }^{2s})+{\beta }^{2s}(1-{\alpha }^{2s}-{\beta }^{2s(n-1)}+{\beta }^{2s(n-1)}{\alpha }^{2s})\hfill \\ \hfill & =({\alpha }^{2s}+{\beta }^{2s})-2{\alpha }^{2s}{\beta }^{2s}-({\alpha }^{2sn}+{\beta }^{2sn})+{\alpha }^{2s}{\beta }^{2s}({\alpha }^{2s(n-1)}+{\beta }^{2s(n-1)}).\hfill \end{array}$$

Using ${\alpha }^{2s}+{\beta }^{2s}=L_{2s}$, ${\alpha }^{2s}{\beta }^{2s}=1$, ${\alpha }^{2sn}+{\beta }^{2sn}=L_{2sn}$, and ${\alpha }^{2s(n-1)}+{\beta }^{2s(n-1)}=L_{2s(n-1)}$, the numerator becomes:

$$L_{2s}-2-L_{2sn}+L_{2s(n-1)}.$$

For the denominator, using ${\alpha }^{2s}+{\beta }^{2s}=L_{2s}$ and ${\alpha }^{2s}{\beta }^{2s}=1$:

$$(1-{\alpha }^{2s})(1-{\beta }^{2s})=1-L_{2s}+1=2-L_{2s}.$$

Therefore:

$${\displaystyle \sum _{l=1}^{n-1}}{\alpha }^{2ls}+{\displaystyle \sum _{l=1}^{n-1}}{\beta }^{2ls}={\displaystyle \frac{L_{2s}-2-L_{2sn}+L_{2s(n-1)}}{2-L_{2s}}}={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{L_{2s}-2}}.$$

Moreover, noting that ${({\alpha }^s-{\beta }^s)}^2=L_{2s}-2{(-1)}^s$, we obtain:

$${\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)\left(L_{2s}-2{(-1)}^s\right)}}-{\displaystyle \frac{2\Omega (n,s)}{L_{2s}-2{(-1)}^s}}.$$

We now substitute the values of $\Omega (n,s)$ from Lemma 4 and distinguish three cases.

Case 1: s even. In this case ${(-1)}^s=1$; so, $L_{2s}-2{(-1)}^s=L_{2s}-2$, and $\Omega (n,s)=n-1$. Therefore:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{{(L_{2s}-2)}^2}}-{\displaystyle \frac{2(n-1)}{L_{2s}-2}}\hfill \\ \hfill & ={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}-2(n-1)(L_{2s}-2)}{{(L_{2s}-2)}^2}}.\hfill \end{array}$$

Simplifying the numerator:

$$\begin{array}{cc}\hfill & 2-L_{2s}+L_{2sn}-L_{2s(n-1)}-2(n-1)L_{2s}+4(n-1)\hfill \\ \hfill & =L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s}),\hfill \end{array}$$

which gives

$${\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2={\displaystyle \frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}}.$$

Case 2: s odd, n even. Here ${(-1)}^s=-1$; so, $L_{2s}-2{(-1)}^s=L_{2s}+2$, and $\Omega (n,s)=-1$. Therefore:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}}+{\displaystyle \frac{2}{L_{2s}+2}}\hfill \\ \hfill & ={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}+2(L_{2s}-2)}{(L_{2s}-2)(L_{2s}+2)}}\hfill \\ \hfill & ={\displaystyle \frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}}.\hfill \end{array}$$

Case 3: s odd, n odd. Here ${(-1)}^s=-1$; so, $L_{2s}-2{(-1)}^s=L_{2s}+2$, and $\Omega (n,s)=0$. Therefore:

$${\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2={\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}}.$$

Combining the three cases yields the stated result. This completes the proof. □

Theorem 3.

Let $n,s\in \mathbb{N}$ with $n\ge 2$, and let ${\left\{F_n^{\left(s\right)}\right\}}_{n\ge 0}$ denote the higher-order Fibonacci numbers of order s. Then, the following summation formula holds:

$${\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2=\left\{\begin{array}{cc}\Phi (s,n),\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ \Psi (s,n),\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ {\rm Y}(s,n),\hfill & \mathit{if}s\mathrm{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd},\hfill \end{array}\right.$$

(16)

where

$$\begin{array}{cc}\hfill \Phi (s,n)& ={\displaystyle \frac{2L_{2s}+(n-1)L_{2s(n+1)}-4+(3n-1)L_{2s(n-1)}+(2-3n)L_{2sn}-n{L}_{2s(n-2)}}{{(L_{2s}-2)}^3}}\hfill \\ \hfill & -{\displaystyle \frac{(n-1)n}{L_{2s}-2}},\hfill \\ \hfill \Psi (s,n)& ={\displaystyle \frac{2L_{2s}+(n-1)L_{2s(n+1)}-4+(3n-1)L_{2s(n-1)}+(2-3n)L_{2sn}-n{L}_{2s(n-2)}}{{(L_{2s}-2)}^2(L_{2s}+2)}}\hfill \\ \hfill & +{\displaystyle \frac{n}{L_{2s}+2}},\hfill \\ \hfill {\rm Y}(s,n)& ={\displaystyle \frac{2L_{2s}+(n-1)L_{2s(n+1)}-4+(3n-1)L_{2s(n-1)}+(2-3n)L_{2sn}-n{L}_{2s(n-2)}}{{(L_{2s}-2)}^2(L_{2s}+2)}}\hfill \\ \hfill & -{\displaystyle \frac{n-1}{L_{2s}+2}}.\hfill \end{array}$$

Here, $L_k$ denotes the kth Lucas number.

Proof. 

Using Equation (6), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \sum _{l=1}^{n-1}}l{\left({\displaystyle \frac{{\alpha }^{ls}-{\beta }^{ls}}{{\alpha }^s-{\beta }^s}}\right)}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{{({\alpha }^s-{\beta }^s)}^2}}\left({\displaystyle \sum _{l=1}^{n-1}}l{\left({\alpha }^{2s}\right)}^l-2{\displaystyle \sum _{l=1}^{n-1}}l{(-1)}^{ls}+{\displaystyle \sum _{l=1}^{n-1}}l{\left({\beta }^{2s}\right)}^l\right).\hfill \end{array}$$

Since $\alpha \beta =-1$, the middle term involves the quantity $\Gamma (n,s)$ defined in Lemma 5. Moreover, setting $\gamma ={\alpha }^{2s}$ and $\delta ={\beta }^{2s}$, Lemma 6 yields

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \frac{1}{{({\alpha }^s-{\beta }^s)}^2}}\left({\displaystyle \sum _{l=1}^{n-1}}l{\gamma }^l-2\Gamma (n,s)+{\displaystyle \sum _{l=1}^{n-1}}l{\delta }^l\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{{({\alpha }^s-{\beta }^s)}^2}}\left[\gamma {\left({\displaystyle \frac{\gamma -{\gamma }^n}{1-\gamma }}\right)}^{\prime }+\delta {\left({\displaystyle \frac{\delta -{\delta }^n}{1-\delta }}\right)}^{\prime }-2\Gamma (n,s)\right],\hfill \end{array}$$

where the prime denotes differentiation with respect to the corresponding variable. Evaluating the derivatives, we obtain

$$\begin{array}{c}\hfill {\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2={\displaystyle \frac{1}{{({\alpha }^s-{\beta }^s)}^2}}\left[{\displaystyle \frac{\gamma -n{\gamma }^n+(n-1){\gamma }^{n+1}}{{(1-\gamma )}^2}}+{\displaystyle \frac{\delta -n{\delta }^n+(n-1){\delta }^{n+1}}{{(1-\delta )}^2}}-2\Gamma (n,s)\right].\end{array}$$

We now combine the two fractions over the common denominator ${(1-\gamma )}^2{(1-\delta )}^2$. Since $\gamma \delta ={\alpha }^{2s}{\beta }^{2s}={\left(\alpha \beta \right)}^{2s}=1$, we have

$$(1-\gamma )(1-\delta )=1-(\gamma +\delta )+\gamma \delta =2-L_{2s},$$

and hence ${(1-\gamma )}^2{(1-\delta )}^2={(L_{2s}-2)}^2$. The combined numerator is

$$\begin{array}{cc}\hfill N& =\left[\gamma -n{\gamma }^n+(n-1){\gamma }^{n+1}\right]{(1-\delta )}^2+\left[\delta -n{\delta }^n+(n-1){\delta }^{n+1}\right]{(1-\gamma )}^2.\hfill \end{array}$$

We expand each group using $\gamma \delta =1$, which gives $\gamma {\delta }^k={\delta }^{k-1}$ and $\delta {\gamma }^k={\gamma }^{k-1}$:

$$\begin{array}{cc}\hfill \gamma (1-2\delta +{\delta }^2)& =\gamma -2+\delta ,\hfill \\ \hfill \delta (1-2\gamma +{\gamma }^2)& =\delta -2+\gamma ,\hfill \\ \hfill -n{\gamma }^n(1-2\delta +{\delta }^2)& =-n{\gamma }^n+2n{\gamma }^{n-1}-n{\gamma }^{n-2},\hfill \\ \hfill -n{\delta }^n(1-2\gamma +{\gamma }^2)& =-n{\delta }^n+2n{\delta }^{n-1}-n{\delta }^{n-2},\hfill \\ \hfill (n-1){\gamma }^{n+1}(1-2\delta +{\delta }^2)& =(n-1){\gamma }^{n+1}-2(n-1){\gamma }^n+(n-1){\gamma }^{n-1},\hfill \\ \hfill (n-1){\delta }^{n+1}(1-2\gamma +{\gamma }^2)& =(n-1){\delta }^{n+1}-2(n-1){\delta }^n+(n-1){\delta }^{n-1}.\hfill \end{array}$$

Summing all terms and using ${\gamma }^k+{\delta }^k=L_{2sk}$ gives

$$\begin{array}{cc}\hfill N& =2L_{2s}-4+(n-1)L_{2s(n+1)}+(2-3n)L_{2sn}+(3n-1)L_{2s(n-1)}-n{L}_{2s(n-2)}.\hfill \end{array}$$

Moreover, since ${({\alpha }^s-{\beta }^s)}^2=L_{2s}-2{(-1)}^s$, we obtain

$${\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2={\displaystyle \frac{N}{(L_{2s}-2{(-1)}^s){(L_{2s}-2)}^2}}-{\displaystyle \frac{2\Gamma (n,s)}{L_{2s}-2{(-1)}^s}}.$$

We now substitute the values of $\Gamma (n,s)$ from Lemma 5 and distinguish three cases.

Case 1: s even. Here ${(-1)}^s=1$; so, $L_{2s}-2{(-1)}^s=L_{2s}-2$, and $\Gamma (n,s)={\displaystyle \frac{n(n-1)}{2}}$. Therefore:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \frac{N}{{(L_{2s}-2)}^3}}-{\displaystyle \frac{n(n-1)}{L_{2s}-2}}=\Phi (s,n).\hfill \end{array}$$

Case 2: s odd, n even. Here ${(-1)}^s=-1$; so, $L_{2s}-2{(-1)}^s=L_{2s}+2$, and $\Gamma (n,s)=-{\displaystyle \frac{n}{2}}$. Therefore:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \frac{N}{{(L_{2s}-2)}^2(L_{2s}+2)}}+{\displaystyle \frac{n}{L_{2s}+2}}=\Psi (s,n).\hfill \end{array}$$

Case 3: s odd, n odd. Here ${(-1)}^s=-1$; so, $L_{2s}-2{(-1)}^s=L_{2s}+2$, and $\Gamma (n,s)={\displaystyle \frac{n-1}{2}}$. Therefore:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2& ={\displaystyle \frac{N}{{(L_{2s}-2)}^2(L_{2s}+2)}}-{\displaystyle \frac{n-1}{L_{2s}+2}}={\rm Y}(s,n).\hfill \end{array}$$

Combining the three cases yields the stated formulas $\Phi (s,n)$, $\Psi (s,n)$, and ${\rm Y}(s,n)$. This completes the proof. □

4. Spectral Properties and Matrix Norms

In this section, we examine the spectral properties and various matrix norms of the r-circulant matrix. Initially, explicit expressions for the eigenvalues are derived by leveraging the r-circulant structure of the matrix, in conjunction with the Binet representation of higher-order Fibonacci numbers. These eigenvalue formulas are subsequently employed to obtain a closed-form expression for the determinant. Thereafter, explicit formulas for the 1-norm and ∞-norm are presented, followed by a comprehensive analysis of the Euclidean norm. Particular emphasis is placed on the spectral norm, for which the lower and upper bounds are established. Given that the behavior of the spectral norm is significantly influenced by the magnitude of the parameter r, the cases $\left|r\right|\ge 1$ and $\left|r\right|<1$ are addressed separately.

Theorem 4.

Let $F_r={Circ}_r\left(F_0^{\left(s\right)},F_1^{\left(s\right)},\dots ,F_{n-1}^{\left(s\right)}\right)$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s, let $\omega =e^{2\pi i/n}$ denote the primitive nth root of unity, and let ρ be a fixed nth root of r, i.e., ${\rho }^n=r$. Note that any two such choices of ρ differ by a factor ${\omega }^j$; so, the set $\{\rho {\omega }^{-k}:0\le k\le n-1\}$ is independent of the choice of ρ. Then, the eigenvalues of $F_r$ are given by

$${\lambda }_k\left(r\right)={\displaystyle \frac{\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)\rho {\omega }^{-k}-r{F}_n^{\left(s\right)}}{\left(1-{\alpha }^s\rho {\omega }^{-k}\right)\left(1-{\beta }^s\rho {\omega }^{-k}\right)}},k=0,1,\dots ,n-1,$$

(17)

where α and β are the roots of $x^2-x-1=0$.

Proof. 

Recall that $\alpha$ and $\beta$ are the roots of the classical equation $x^2-x-1=0$, whereas the sequence $\left\{F_n^{\left(s\right)}\right\}$ satisfies a second-order recurrence whose characteristic roots are ${\alpha }^s$ and ${\beta }^s$, the roots of $x^2-L_{s}x+{(-1)}^s=0$. This distinction is used throughout the proof.

Using Equation (1), the eigenvalues of the r-circulant matrix $F_r$ are given by

$${\lambda }_k\left(r\right)={\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}{\left(\rho {\omega }^{-k}\right)}^l,k=0,1,\dots ,n-1.$$

Employing the Binet-type formula for the higher-order Fibonacci numbers,

$$F_l^{\left(s\right)}={\displaystyle \frac{{\alpha }^{ls}-{\beta }^{ls}}{{\alpha }^s-{\beta }^s}},$$

we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}{\left(\rho {\omega }^{-k}\right)}^l& ={\displaystyle \sum _{l=0}^{n-1}}{\displaystyle \frac{{\alpha }^{ls}-{\beta }^{ls}}{{\alpha }^s-{\beta }^s}}{\left(\rho {\omega }^{-k}\right)}^l\hfill \\ \hfill & ={\displaystyle \frac{1}{{\alpha }^s-{\beta }^s}}\left({\displaystyle \sum _{l=0}^{n-1}}{\left({\alpha }^s\rho {\omega }^{-k}\right)}^l-{\displaystyle \sum _{l=0}^{n-1}}{\left({\beta }^s\rho {\omega }^{-k}\right)}^l\right).\hfill \end{array}$$

Since both sums are geometric series, this yields

$${\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}{\left(\rho {\omega }^{-k}\right)}^l={\displaystyle \frac{1}{{\alpha }^s-{\beta }^s}}\left[{\displaystyle \frac{1-{\left({\alpha }^s\rho {\omega }^{-k}\right)}^n}{1-{\alpha }^s\rho {\omega }^{-k}}}-{\displaystyle \frac{1-{\left({\beta }^s\rho {\omega }^{-k}\right)}^n}{1-{\beta }^s\rho {\omega }^{-k}}}\right].$$

Since ${\rho }^n=r$, we have ${\left(\rho {\omega }^{-k}\right)}^n={\rho }^n{\omega }^{-kn}=r·1=r$, and hence

$$\begin{array}{cc}\hfill {\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}{\left(\rho {\omega }^{-k}\right)}^l& ={\displaystyle \frac{1}{{\alpha }^s-{\beta }^s}}\left[{\displaystyle \frac{1-r{\alpha }^{sn}}{1-{\alpha }^s\rho {\omega }^{-k}}}-{\displaystyle \frac{1-r{\beta }^{sn}}{1-{\beta }^s\rho {\omega }^{-k}}}\right]\hfill \\ \hfill & ={\displaystyle \frac{r({\beta }^{sn}-{\alpha }^{sn})+\rho {\omega }^{-k}({\alpha }^s-{\beta }^s)+r\rho {\omega }^{-k}({\alpha }^{sn}{\beta }^s-{\beta }^{sn}{\alpha }^s)}{({\alpha }^s-{\beta }^s)(1-{\alpha }^s\rho {\omega }^{-k})(1-{\beta }^s\rho {\omega }^{-k})}}.\hfill \end{array}$$

Since $\alpha \beta =-1$, we have ${\alpha }^{sn}{\beta }^s-{\beta }^{sn}{\alpha }^s={(-1)}^s({\alpha }^{s(n-1)}-{\beta }^{s(n-1)})$. Therefore, by the Binet formulas for $F_n^{\left(s\right)}$ and $F_{n-1}^{\left(s\right)}$, the above expression reduces to

$${\displaystyle \sum _{l=0}^{n-1}}{F}_l^{\left(s\right)}{\left(\rho {\omega }^{-k}\right)}^l={\displaystyle \frac{-r{F}_n^{\left(s\right)}+\rho {\omega }^{-k}+{(-1)}^{s}r\rho {\omega }^{-k}{F}_{n-1}^{\left(s\right)}}{(1-{\alpha }^s\rho {\omega }^{-k})(1-{\beta }^s\rho {\omega }^{-k})}}.$$

Equivalently,

$${\lambda }_k\left(r\right)={\displaystyle \frac{\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)\rho {\omega }^{-k}-r{F}_n^{\left(s\right)}}{(1-{\alpha }^s\rho {\omega }^{-k})(1-{\beta }^s\rho {\omega }^{-k})}},$$

which completes the proof. □

Theorem 5.

Let $F_r$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s. Assume that $r\ne {\alpha }^{-sn}$ and $r\ne {\beta }^{-sn}$. Then, the determinant of $F_r$ is given by

$$det\left(F_r\right)={\displaystyle \frac{{(-1)}^n\left[{\left(r{F}_n^{\left(s\right)}\right)}^n-r{\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)}^n\right]}{1-r{L}_{sn}+r^2{(-1)}^{sn}}},$$

(18)

where the denominator $1-r{L}_{sn}+r^2{(-1)}^{sn}=(1-r{\alpha }^{sn})(1-r{\beta }^{sn})$ is nonzero under the stated assumption.

Proof. 

It is well known that the determinant of a square matrix equals the product of its eigenvalues. Hence, by Theorem 4, we have

$$det\left(F_r\right)={\displaystyle \prod _{k=0}^{n-1}}{\lambda }_k\left(r\right).$$

Substituting the explicit expression of ${\lambda }_k\left(r\right)$ yields

$$\begin{array}{cc}\hfill det\left(F_r\right)& ={\displaystyle \prod _{k=0}^{n-1}}{\displaystyle \frac{\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)\rho {\omega }^{-k}-r{F}_n^{\left(s\right)}}{(1-{\alpha }^s\rho {\omega }^{-k})(1-{\beta }^s\rho {\omega }^{-k})}}.\hfill \end{array}$$

Rearranging the numerator terms, we obtain

$$\begin{array}{cc}\hfill det\left(F_r\right)& ={(-1)}^n{\displaystyle \prod _{k=0}^{n-1}}{\displaystyle \frac{r{F}_n^{\left(s\right)}-\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)\rho {\omega }^{-k}}{(1-{\alpha }^s\rho {\omega }^{-k})(1-{\beta }^s\rho {\omega }^{-k})}}.\hfill \end{array}$$

Using the identity

$${\displaystyle \prod _{k=0}^{n-1}}(x-y{\omega }^{-k})=x^n-y^n,$$

together with ${\left(\rho {\omega }^{-k}\right)}^n=r$, we arrive at

$$det\left(F_r\right)={(-1)}^n{\displaystyle \frac{{\left(r{F}_n^{\left(s\right)}\right)}^n-r{\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)}^n}{(1-r{\alpha }^{sn})(1-r{\beta }^{sn})}}.$$

Finally, since ${\alpha }^{sn}+{\beta }^{sn}=L_{sn}$ and ${\left(\alpha \beta \right)}^{sn}={(-1)}^{sn}$, the denominator simplifies to

$$(1-r{\alpha }^{sn})(1-r{\beta }^{sn})=1-r{L}_{sn}+r^2{(-1)}^{sn}.$$

Therefore:

$$det\left(F_r\right)={(-1)}^n{\displaystyle \frac{{\left(r{F}_n^{\left(s\right)}\right)}^n-r{\left(1+{(-1)}^{s}r{F}_{n-1}^{\left(s\right)}\right)}^n}{1-r{L}_{sn}+r^2{(-1)}^{sn}}},$$

which completes the proof. □

Remark 1.

When $r=0$, the matrix $F_r$ reduces to a strictly upper triangular matrix with zero diagonal and with $F_0^{\left(s\right)}=0$ on all diagonal entries. In this case, all eigenvalues are zero and hence $det\left(F_0\right)=0$.

The following lemma is obvious.

Lemma 7.

Let $F_r$ be an r-circulant square matrix. Then, its induced 1-norm and ∞-norm coincide, that is,

$$\parallel F_r{\parallel }_1={\parallel F_r\parallel }_{\infty }.$$

Theorem 6.

Let $s\ne 0$ and let $F_r$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s. Then, the induced 1-norm and ∞-norm of $F_r$ coincide and are given by

$$\parallel F_r{\parallel }_1={\parallel F_r\parallel }_{\infty }=\left\{\begin{array}{cc}{\displaystyle \left|r\right|\frac{1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}}{1-L_s+{(-1)}^s},}\hfill & \left|r\right|\ge 1,\hfill \\ {\displaystyle \frac{1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}}{1-L_s+{(-1)}^s},}\hfill & \left|r\right|<1.\hfill \end{array}\right.$$

(19)

Here, $L_s$ denotes the sth Lucas number.

Proof. 

By Lemma 7, we have $\parallel F_r{\parallel }_1={\parallel F_r\parallel }_{\infty }$; hence it suffices to determine the maximum absolute row sum of $F_r$.

Since $F_l^{\left(s\right)}\ge 0$ for all $0\le l\le n-1$, every absolute value below equals the corresponding term, and we may write the sums without bars after this point. Using $F_0^{\left(s\right)}=0$, the i-th row of $F_r$ consists of the $n-i+1$ entries $F_0^{\left(s\right)},F_1^{\left(s\right)},\dots ,F_{n-i}^{\left(s\right)}$ carrying no factor of r, together with the $i-1$ wrap-around entries $r{F}_{n-i+1}^{\left(s\right)},\dots ,r{F}_{n-1}^{\left(s\right)}$ carrying the factor r. Hence its absolute row sum is

$$S_i={\displaystyle \sum _{l=0}^{n-i}}{F}_l^{\left(s\right)}+\left|r\right|{\displaystyle \sum _{l=n-i+1}^{n-1}}{F}_l^{\left(s\right)}={\displaystyle \sum _{l=1}^{n-i}}{F}_l^{\left(s\right)}+\left|r\right|{\displaystyle \sum _{l=n-i+1}^{n-1}}{F}_l^{\left(s\right)},1\le i\le n,$$

where the last step uses $F_0^{\left(s\right)}=0$. In particular,

$$S_1={\displaystyle \sum _{l=1}^{n-1}}{F}_l^{\left(s\right)},S_n=\left|r\right|{\displaystyle \sum _{l=1}^{n-1}}{F}_l^{\left(s\right)},$$

so row 1 carries no factor of r, whereas row n has every off-diagonal entry multiplied by r. Note that the two index ranges $\{1,\dots ,n-i\}$ and $\{n-i+1,\dots ,n-1\}$ partition $\{1,\dots ,n-1\}$, a fact we use repeatedly below.

• Case 1: $\left|r\right|\ge 1$. For any $2\le i\le n-1$, we have

$$S_i={\displaystyle \sum _{l=1}^{n-i}}{F}_l^{\left(s\right)}+\left|r\right|{\displaystyle \sum _{l=n-i+1}^{n-1}}{F}_l^{\left(s\right)}\le \left|r\right|{\displaystyle \sum _{l=1}^{n-i}}{F}_l^{\left(s\right)}+\left|r\right|{\displaystyle \sum _{l=n-i+1}^{n-1}}{F}_l^{\left(s\right)}=\left|r\right|{\displaystyle \sum _{l=1}^{n-1}}{F}_l^{\left(s\right)}=S_n.$$

Likewise $S_1={\sum }_{l=1}^{n-1}{F}_l^{\left(s\right)}\le \left|r\right|{\sum }_{l=1}^{n-1}{F}_l^{\left(s\right)}=S_n$. Thus the maximum row sum is attained by row n, and

$$\parallel F_r{\parallel }_1=\parallel F_r{\parallel }_{\infty }=\left|r\right|{\displaystyle \sum _{l=1}^{n-1}}{F}_l^{\left(s\right)}.$$

Applying the summation formula of Theorem 1 gives

$$\parallel F_r{\parallel }_1=\left|r\right|{\displaystyle \frac{1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}}{1-L_s+{(-1)}^s}}.$$

Case 2: $\left|r\right|<1$. For any $2\le i\le n-1$, we get

$$S_i={\displaystyle \sum _{l=1}^{n-i}}{F}_l^{\left(s\right)}+\left|r\right|{\displaystyle \sum _{l=n-i+1}^{n-1}}{F}_l^{\left(s\right)}\le {\displaystyle \sum _{l=1}^{n-i}}{F}_l^{\left(s\right)}+{\displaystyle \sum _{l=n-i+1}^{n-1}}{F}_l^{\left(s\right)}={\displaystyle \sum _{l=1}^{n-1}}{F}_l^{\left(s\right)}=S_1.$$

Likewise $S_n=\left|r\right|{\sum }_{l=1}^{n-1}{F}_l^{\left(s\right)}\le {\sum }_{l=1}^{n-1}{F}_l^{\left(s\right)}=S_1$. Thus the maximum row sum is attained by row 1, and

$$\parallel F_r{\parallel }_1={\parallel F_r\parallel }_{\infty }={\displaystyle \sum _{l=1}^{n-1}}{F}_l^{\left(s\right)}.$$

Applying Theorem 1 once more yields

$$\parallel F_r{\parallel }_1={\displaystyle \frac{1-F_n^{\left(s\right)}+{(-1)}^s{F}_{n-1}^{\left(s\right)}}{1-L_s+{(-1)}^s}}.$$

Combining the two cases completes the proof. □

Theorem 7.

Let $F_r$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s. Then, the Euclidean (Frobenius) norm of $F_r$ is given by

$$\parallel F_r{\parallel }_E=\left\{\begin{array}{cc}{\displaystyle \sqrt{n\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}+{\left(\right|r|}^2-1)\Phi (s,n)},}\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sqrt{n\frac{L_{2sn}-L_{2s(n-1)}+L_{2s}-2}{(L_{2s}-2)(L_{2s}+2)}+{\left(\right|r|}^2-1)\Psi (s,n)},}\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sqrt{n\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}+{\left(\right|r|}^2-1){\rm Y}(s,n)},}\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(20)

Here, $\Phi (s,n)$, $\Psi (s,n)$, and ${\rm Y}(s,n)$ are defined as in Theorem 3, and $L_k$ denotes the kth Lucas number.

Proof. 

By definition, the Euclidean (Frobenius) norm of $F_r$ is given by

$$\parallel F_r{\parallel }_E=\sqrt{\sum _{i=1}^n\sum _{j=1}^n{\left|f_{ij}\right|}^2}.$$

We first analyze the structure of $F_r$ to count how many times each entry $F_l^{\left(s\right)}$ appears in the matrix. The r-circulant matrix $F_r={Circ}_r(F_0^{\left(s\right)},F_1^{\left(s\right)},\dots ,F_{n-1}^{\left(s\right)})$ has the property that the entry $F_l^{\left(s\right)}$ appears exactly:

• $(n-l)$ times without the factor r (in positions above the main diagonal and on it), contributing $(n-l)|F_l^{\left(s\right)}{|}^2$ to $\parallel F_r{\parallel }_E^2$,
• l times multiplied by r (in positions below the main diagonal), contributing ${l\left|r\right|}^2{\left|F_l^{\left(s\right)}\right|}^2$ to $\parallel F_r{\parallel }_E^2$.

Therefore:

$$\parallel F_r{\parallel }_E^2={\displaystyle \sum _{l=0}^{n-1}}\left[(n-l)|F_l^{\left(s\right)}{|}^2+{l\left|r\right|}^2{\left|F_l^{\left(s\right)}\right|}^2\right]={\displaystyle \sum _{l=0}^{n-1}}\left[(n-l)+{l\left|r\right|}^2\right]{\left(F_l^{\left(s\right)}\right)}^2.$$

Taking into account the structure of the r-circulant matrix $F_r$, each entry $F_l^{\left(s\right)}$ appears $(n-l)$ times without the factor r and l times multiplied by r. Hence, we obtain

$$\begin{array}{cc}\hfill \parallel F_r{\parallel }_E& =\sqrt{n{\left(F_0^{\left(s\right)}\right)}^2+\sum _{l=1}^{n-1}\left[(n-l)+{l\left|r\right|}^2\right]{\left(F_l^{\left(s\right)}\right)}^2}.\hfill \end{array}$$

Since $F_0^{\left(s\right)}=0$, this reduces to

$$\begin{array}{cc}\hfill \parallel F_r{\parallel }_E& =\sqrt{\sum _{l=1}^{n-1}\left[n-l(1-|r{|}^2)\right]{\left(F_l^{\left(s\right)}\right)}^2}\hfill \\ \hfill & =\sqrt{n\sum _{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2+{\left(\right|r|}^2-1)\sum _{l=1}^{n-1}l{\left(F_l^{\left(s\right)}\right)}^2}.\hfill \end{array}$$

We now substitute the explicit expressions from Theorems 2 and 3 and distinguish three cases.

Case 1: s even. Substituting from Theorems 2 and 3:

$$n{\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2=n·{\displaystyle \frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},$$

$${\left(\right|r|}^2-1){\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2={\left(\right|r|}^2-1)\Phi (s,n).$$

Therefore:

$$\parallel F_r{\parallel }_E=\sqrt{{\displaystyle \frac{n\left[L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})\right]}{{(L_{2s}-2)}^2}}+{\left(\right|r|}^2-1)\Phi (s,n)}.$$

Case 2: s odd, n even. Substituting from Theorems 2 and 3:

$$n{\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2=n·{\displaystyle \frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},$$

$${\left(\right|r|}^2-1){\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2={\left(\right|r|}^2-1)\Psi (s,n).$$

Therefore:

$$\parallel F_r{\parallel }_E=\sqrt{{\displaystyle \frac{n\left[L_{2s}+L_{2sn}-L_{2s(n-1)}-2\right]}{(L_{2s}-2)(L_{2s}+2)}}+{\left(\right|r|}^2-1)\Psi (s,n)}.$$

Case 3: s odd, n odd. Substituting from Theorems 2 and 3:

$$n{\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2=n·{\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},$$

$${\left(\right|r|}^2-1){\displaystyle \sum _{l=1}^{n-1}}l{\left(F_l^{\left(s\right)}\right)}^2={\left(\right|r|}^2-1){\rm Y}(s,n).$$

Therefore:

$$\parallel F_r{\parallel }_E=\sqrt{{\displaystyle \frac{n\left[2-L_{2s}+L_{2sn}-L_{2s(n-1)}\right]}{(L_{2s}-2)(L_{2s}+2)}}+{\left(\right|r|}^2-1){\rm Y}(s,n)}.$$

Combining the three cases yields the desired result. This completes the proof. □

Theorem 8.

Let $\left|r\right|\ge 1$ and let $F_r$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s. Then, the spectral norm $\parallel F_r{\parallel }_2$ admits the following lower bounds:

$$\parallel F_r{\parallel }_2\ge \left\{\begin{array}{cc}{\displaystyle \sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(21)

Here, $L_k$ denotes the kth Lucas number.

Proof. 

From the definition of the Euclidean (Frobenius) norm, we have

$$\parallel F_r{\parallel }_E^2=n{\left(F_0^{\left(s\right)}\right)}^2+{\displaystyle \sum _{l=1}^{n-1}}\left[(n-l)+{l\left|r\right|}^2\right]{\left(F_l^{\left(s\right)}\right)}^2.$$

Since $F_0^{\left(s\right)}=0$, this reduces to

$$\parallel F_r{\parallel }_E^2={\displaystyle \sum _{l=1}^{n-1}}\left[(n-l)+{l\left|r\right|}^2\right]{\left(F_l^{\left(s\right)}\right)}^2.$$

For $\left|r\right|\ge 1$, we clearly have $(n-l)+{l\left|r\right|}^2\ge n$ for all $l=1,\dots ,n-1$. Therefore:

$$\parallel F_r{\parallel }_E^2\ge n{\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2.$$

Taking square roots yields

$${\displaystyle \frac{\parallel F_r{\parallel }_E}{\sqrt{n}}}\ge \sqrt{\sum _{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2}.$$

Using the explicit expressions for ${\sum }_{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2$ given in Theorem 2, we obtain

$${\displaystyle \frac{\parallel F_r{\parallel }_E}{\sqrt{n}}}\ge \left\{\begin{array}{cc}{\displaystyle \sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Finally, since the spectral norm is bounded below by the Euclidean norm via

$$\parallel F_r{\parallel }_2\ge {\displaystyle \frac{1}{\sqrt{n}}}{\parallel F_r\parallel }_E,$$

the desired lower bounds for $\parallel F_r{\parallel }_2$ follow immediately. This completes the proof. □

Theorem 9.

Let $\left|r\right|\ge 1$ and let $F_r$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s. Then, the spectral norm $\parallel F_r{\parallel }_2$ satisfies the following upper bounds:

$$\begin{array}{cc}\hfill \parallel F_r{\parallel }_2& \le \left|r\right|\sqrt{{\displaystyle \frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}}+{\left({\displaystyle \frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}}\right)}^2},\hfill \\ \hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ \hfill \parallel F_r{\parallel }_2& \le \left|r\right|\sqrt{{\displaystyle \frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}}+{\left({\displaystyle \frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}}\right)}^2},\hfill \\ \hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ \hfill \parallel F_r{\parallel }_2& \le \left|r\right|\sqrt{{\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}}+{\left({\displaystyle \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}}\right)}^2},\hfill \\ \hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}$$

(22)

Here, $L_k$ denotes the kth Lucas number.

Proof. 

Define the matrices

$$S=\left(\begin{array}{cccccc}{F}_0^{\left(s\right)}& 1& 1& \cdots & 1& 1\\ r{F}_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}& 1& \cdots & 1& 1\\ r{F}_{n-2}^{\left(s\right)}& r{F}_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}& \cdots & 1& 1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ r{F}_2^{\left(s\right)}& r{F}_3^{\left(s\right)}& r{F}_4^{\left(s\right)}& \cdots & F_0^{\left(s\right)}& 1\\ r{F}_1^{\left(s\right)}& r{F}_2^{\left(s\right)}& r{F}_3^{\left(s\right)}& \cdots & r{F}_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}\end{array}\right),$$

and

$$T=\left(\begin{array}{cccccc}1& F_1^{\left(s\right)}& F_2^{\left(s\right)}& \cdots & F_{n-2}^{\left(s\right)}& F_{n-1}^{\left(s\right)}\\ 1& 1& F_1^{\left(s\right)}& \cdots & F_{n-3}^{\left(s\right)}& F_{n-2}^{\left(s\right)}\\ 1& 1& 1& \cdots & F_{n-4}^{\left(s\right)}& F_{n-3}^{\left(s\right)}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 1& 1& 1& \cdots & 1& F_1^{\left(s\right)}\\ 1& 1& 1& \cdots & 1& 1\end{array}\right).$$

Then, the matrix $F_r$ can be written as the Hadamard (entrywise) product

$$F_r=S\circ T.$$

Let

$$\mu \left(S\right)=\underset{1\le i\le n}{max}{\left({\displaystyle \sum _{j=1}^n}{\left|s_{ij}\right|}^2\right)}^{1/2}\mathrm{and}\nu \left(T\right)=\underset{1\le j\le n}{max}{\left({\displaystyle \sum _{i=1}^n}{\left|t_{ij}\right|}^2\right)}^{1/2}.$$

A direct computation yields

$$\mu \left(S\right)=\sqrt{{\left(F_0^{\left(s\right)}\right)}^2+{\left|r\right|}^2\sum _{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2}=\left|r\right|\sqrt{\sum _{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2},$$

since $F_0^{\left(s\right)}=0$. Using the explicit expressions for ${\sum }_{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2$ given in Theorem 2, we obtain

$$\mu \left(S\right)=\left\{\begin{array}{cc}{\displaystyle \left|r\right|\sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \left|r\right|\sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \left|r\right|\sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

Similarly,

$$\nu \left(T\right)=\sqrt{1+\sum _{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2},$$

which, again by Theorem 2, gives

$$\nu \left(T\right)=\left\{\begin{array}{cc}{\displaystyle \sqrt{1+\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \sqrt{1+\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \sqrt{1+\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

By the well-known inequality for the Hadamard product,

$${\parallel S\circ T\parallel }_2\le \mu \left(S\right)\nu \left(T\right),$$

we conclude that $\parallel F_r{\parallel }_2\le \mu \left(S\right)\nu \left(T\right)$. Combining the above expressions for $\mu \left(S\right)$ and $\nu \left(T\right)$ yields exactly the stated upper bounds for the spectral norm of $F_r$. This completes the proof. □

Theorem 10.

Let $F_r$ be an $n\times n$ r-circulant matrix generated by higher-order Fibonacci numbers. For $\left|r\right|<1$, the spectral norm of $F_r$ satisfies the following lower bounds:

$${\displaystyle \left\{{\displaystyle \begin{array}{ll}{\displaystyle \left|r\right|\sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}}\le {\parallel F_r\parallel }_2,}\hfill & s\mathit{even},\hfill \\ \left|r\right|\sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}}\le {\parallel F_r\parallel }_2,& s\mathit{odd},n\mathit{even},\\ \left|r\right|\sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}}\le {\parallel F_r\parallel }_2,& s\mathit{odd},n\mathit{odd}.\end{array}}\right.}$$

(23)

Proof. 

From the definition of the Euclidean (Frobenius) norm,

$$\parallel F_r{\parallel }_E^2=n{\left(F_0^{\left(s\right)}\right)}^2+{\displaystyle \sum _{l=1}^{n-1}}\left((n-l)+{l\left|r\right|}^2\right){\left(F_l^{\left(s\right)}\right)}^2.$$

Since $F_0^{\left(s\right)}=0$, we obtain

$$\parallel F_r{\parallel }_E^2={\displaystyle \sum _{l=1}^{n-1}}\left((n-l)+{l\left|r\right|}^2\right){\left(F_l^{\left(s\right)}\right)}^2.$$

Now, since $\left|r\right|<1$, it follows that

$$(n-l)+{l\left|r\right|}^2\ge {(n-l)\left|r\right|}^2+{l\left|r\right|}^2=n{\left|r\right|}^2,$$

and therefore

$$\parallel F_r{\parallel }_E^2\ge n{\left|r\right|}^2{\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2.$$

Using (15), we obtain

$${\displaystyle {\displaystyle \sum _{l=1}^{n-1}}{\left(F_l^{\left(s\right)}\right)}^2=\left\{{\displaystyle \begin{array}{ll}{\displaystyle \frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2},}\hfill & s\mathrm{even},\hfill \\ \frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)},& s\mathrm{odd},n\mathrm{even},\\ \frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)},& s\mathrm{odd},n\mathrm{odd}.\end{array}}\right.}$$

Hence,

$${\displaystyle \frac{\parallel F_r{\parallel }_E}{\sqrt{n}}\ge \left|r\right|\left\{{\displaystyle \begin{array}{ll}{\displaystyle \sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & s\mathrm{even},\hfill \\ \sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},& s\mathrm{odd},n\mathrm{even},\\ \sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},& s\mathrm{odd},n\mathrm{odd}.\end{array}}\right.}$$

Finally, by the inequality (10),

$$\parallel F_r{\parallel }_2\ge {\displaystyle \frac{1}{\sqrt{n}}}{\parallel F_r\parallel }_E,$$

we conclude that

$${\displaystyle \parallel F_r{\parallel }_2\ge \left\{{\displaystyle \begin{array}{ll}{\displaystyle \left|r\right|\sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & s\mathrm{even},\hfill \\ \left|r\right|\sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},& s\mathrm{odd},n\mathrm{even},\\ \left|r\right|\sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},& s\mathrm{odd},n\mathrm{odd},\end{array}}\right.}$$

which completes the proof. □

Theorem 11.

Let $\left|r\right|<1$ and let $F_r$ be the r-circulant matrix generated by the higher-order Fibonacci numbers of order s. Then, the spectral norm $\parallel F_r{\parallel }_2$ admits the following upper bounds:

$$\parallel F_r{\parallel }_2\le \left\{\begin{array}{cc}{\displaystyle \sqrt{n\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & \mathit{if}s\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sqrt{n\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sqrt{n\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathit{if}s\mathit{is}\mathit{odd}\mathit{and}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(24)

Here, $L_k$ denotes the kth Lucas number.

Proof. 

Define the matrices

$$S^{\prime }=\left(\begin{array}{cccccc}1& 1& 1& \cdots & 1& 1\\ r& 1& 1& \cdots & 1& 1\\ r& r& 1& \cdots & 1& 1\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ r& r& r& \cdots & 1& 1\\ r& r& r& \cdots & r& 1\end{array}\right)$$

and

$$T^{\prime }=\left(\begin{array}{cccccc}{F}_0^{\left(s\right)}& F_1^{\left(s\right)}& F_2^{\left(s\right)}& \cdots & F_{n-2}^{\left(s\right)}& F_{n-1}^{\left(s\right)}\\ F_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}& F_1^{\left(s\right)}& \cdots & F_{n-3}^{\left(s\right)}& F_{n-2}^{\left(s\right)}\\ F_{n-2}^{\left(s\right)}& F_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}& \cdots & F_{n-4}^{\left(s\right)}& F_{n-3}^{\left(s\right)}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ F_2^{\left(s\right)}& F_3^{\left(s\right)}& F_4^{\left(s\right)}& \cdots & F_0^{\left(s\right)}& F_1^{\left(s\right)}\\ F_1^{\left(s\right)}& F_2^{\left(s\right)}& F_3^{\left(s\right)}& \cdots & F_{n-1}^{\left(s\right)}& F_0^{\left(s\right)}\end{array}\right).$$

Then, the matrix $F_r$ can be written as the Hadamard product

$$F_r=S^{\prime }\circ T^{\prime }.$$

Let

$$\mu \left(S^{\prime }\right)=\underset{1\le i\le n}{max}{\left({\displaystyle \sum _{j=1}^n}{\left|s_{ij}^{\prime }\right|}^2\right)}^{1/2}\mathrm{and}\nu \left(T^{\prime }\right)=\underset{1\le j\le n}{max}{\left({\displaystyle \sum _{i=1}^n}{\left|t_{ij}^{\prime }\right|}^2\right)}^{1/2}.$$

Since each row of $S^{\prime }$ contains exactly n entries of modulus at most 1 and $\left|r\right|<1$, we immediately obtain

$$\mu \left(S^{\prime }\right)=\sqrt{n}.$$

Moreover,

$$\nu \left(T^{\prime }\right)=\sqrt{\sum _{l=0}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2}.$$

Using the explicit expressions for ${\sum }_{l=1}^{n-1}{\left(F_l^{\left(s\right)}\right)}^2$ given in Theorem 2, we have

$$\nu \left(T^{\prime }\right)=\left\{\begin{array}{cc}{\displaystyle \sqrt{\frac{L_{2sn}-L_{2s(n-1)}+(2n-1)(2-L_{2s})}{{(L_{2s}-2)}^2}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \sqrt{\frac{L_{2s}+L_{2sn}-L_{2s(n-1)}-2}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \sqrt{\frac{2-L_{2s}+L_{2sn}-L_{2s(n-1)}}{(L_{2s}-2)(L_{2s}+2)}},}\hfill & \mathrm{if}s\mathrm{is}\mathrm{odd}\mathrm{and}n\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

By the standard inequality for the Hadamard product,

$$\parallel S^{\prime }\circ T^{\prime }{\parallel }_2\le \mu \left(S^{\prime }\right)\nu \left(T^{\prime }\right),$$

we conclude that

$$\parallel F_r{\parallel }_2\le \sqrt{n}\nu \left(T^{\prime }\right).$$

Substituting the above expressions for $\nu \left(T^{\prime }\right)$ yields exactly the stated upper bounds for the spectral norm of $F_r$ when $\left|r\right|<1$. This completes the proof. □

5. Numerical Examples

In this section, we present numerical examples to illustrate the applicability of the theoretical results obtained in the previous sections. All examples are constructed by selecting specific values of the parameters s, n, and r in the r-circulant matrix

$$F_r=Circr\left(F_0^{\left(s\right)},F_1^{\left(s\right)},\dots ,F_{n-1}^{\left(s\right)}\right),$$

whose entries consist of higher-order Fibonacci numbers.

5.1. Example 1: For s Even and $r=1$

Let $s=2$, $n=4$, and $r=1$. Using the higher-order Fibonacci sequence

$$F_n^{\left(2\right)}=(0,1,3,8,\dots ),$$

the corresponding circulant matrix takes the form

$$F=\left(\begin{array}{cccc}0& 1& 3& 8\\ 8& 0& 1& 3\\ 3& 8& 0& 1\\ 1& 3& 8& 0\end{array}\right).$$

(i)

Matrix norms:

From Theorem 6, the maximum column and row sum norms are

$${\parallel F\parallel }_1={\parallel F\parallel }_{\infty }={\displaystyle \frac{1-F_4^{\left(2\right)}+F_3^{\left(2\right)}}{1-L_2+1}}={\displaystyle \frac{1-21+8}{-1}}=12.$$

Using Theorem 7, the Euclidean norm is computed as

$${\parallel F\parallel }_E=\sqrt{4\left[{\displaystyle \frac{L_{16}-L_{12}+7(2-L_4)}{{(L_4-2)}^2}}\right]}=2\sqrt{74}.$$

(ii)

Spectral norm bounds:

Since s is even, Theorem 8 yields the lower bound

$${\parallel F\parallel }_2\ge \sqrt{{\displaystyle \frac{L_{16}-L_{12}+7(2-L_4)}{{(L_4-2)}^2}}}=\sqrt{74},$$

while Theorem 9 gives the upper bound

$${\parallel F\parallel }_2\le \sqrt{74+{74}^2}=5\sqrt{222}.$$

These results verify the tightness of the theoretical bounds obtained for the case $r=1$. A direct numerical computation via the eigenvalues of F gives ${\parallel F\parallel }_2=12$. Since $\sqrt{74}\approx 8.602$ and $5\sqrt{222}\approx 74.498$, the actual spectral norm lies strictly within the derived interval $[\sqrt{74},5\sqrt{222}]$, confirming the validity of the bounds.

5.2. Example 2: For s Odd and $\left|r\right|>1$

Let $s=3$, $n=4$, and $r=2$. The corresponding r-circulant matrix is

$$F=\left(\begin{array}{cccc}0& 1& 4& 17\\ 34& 0& 1& 4\\ 8& 34& 0& 1\\ 2& 8& 34& 0\end{array}\right).$$

(i)

Matrix norms:

From Theorem 6, we obtain

$${\parallel F\parallel }_1={\parallel F\parallel }_{\infty }=2{\displaystyle \frac{1-F_4^{\left(3\right)}-F_3^{\left(3\right)}}{1-L_3+{(-1)}^3}}=44.$$

Applying Theorem 7, the Euclidean norm becomes

$${\parallel F\parallel }_E=\sqrt{1224+2700}=6\sqrt{109}.$$

(ii)

Spectral norm bounds:

Since s is odd and n is even, Theorem 8 yields

$${\parallel F\parallel }_2\ge \sqrt{{\displaystyle \frac{L_6+L_{24}-L_{18}-2}{(L_6-2)(L_6+2)}}}=3\sqrt{34},$$

whereas Theorem 9 provides the upper bound

$${\parallel F\parallel }_2\le 2\sqrt{306+{306}^2}=2\sqrt{\mathrm{93,942}}=6\sqrt{\mathrm{10,438}}.$$

These numerical values confirm that the spectral norm of $F_r$ is well captured by the obtained theoretical bounds and that the parameter r significantly influences the magnitude of the norm. A direct numerical computation yields ${\parallel F\parallel }_2\approx 40.976$. The theoretical bounds give

$$3\sqrt{34}\approx 17.493\le {\parallel F\parallel }_2\le 6\sqrt{\mathrm{10,438}}\approx 612.999,$$

and indeed $40.976$ lies within this interval, confirming the theoretical results.

5.3. Example 3: For s Odd and $\left|r\right|<1$

Let $s=3$, $n=4$, and $r=\frac{1}{2}$. The corresponding higher-order Fibonacci numbers are

$$F_n^{\left(3\right)}=(0,1,4,17,\dots ).$$

Thus, the r-circulant matrix $F_r$ is given by

$$F_r=\left(\begin{array}{cccc}0& 1& 4& 17\\ {\displaystyle \frac{17}{2}}& 0& 1& 4\\ 2& {\displaystyle \frac{17}{2}}& 0& 1\\ {\displaystyle \frac{1}{2}}& 2& {\displaystyle \frac{17}{2}}& 0\end{array}\right).$$

(i)

Euclidean norm:

Using the explicit formula in Theorem 7, the Euclidean norm of $F_r$ is computed as

$$\parallel F_r{\parallel }_E=\sqrt{4\left[{\displaystyle \frac{L_6+L_{24}-L_{18}-2}{(L_6-2)(L_6+2)}}\right]+\left(\frac{1}{4}-1\right)\Psi (3,4)}=\sqrt{1224-\frac{3}{4}\Psi (3,4)}=3\sqrt{61}.$$

(ii)

Spectral norm bound:

Since $\left|r\right|<1$ and s is odd with even n, Theorem 11 yields the upper bound

$$\parallel F_r{\parallel }_2\le \sqrt{4·{\displaystyle \frac{L_6+L_{24}-L_{18}-2}{(L_6-2)(L_6+2)}}}=6\sqrt{34}.$$

This confirms that, for $\left|r\right|<1$, the spectral norm of $F_r$ is dominated by the unweighted part of the matrix and is independent of r, in full agreement with the theoretical results. A direct numerical computation gives $\parallel F_r{\parallel }_2\approx 17.748$. Since $6\sqrt{34}\approx 34.985$, the bound $\parallel F_r{\parallel }_2\le 6\sqrt{34}$ is indeed satisfied, validating Theorem 11 for this case.

6. Conclusions

In this study, a new class of r-circulant matrices generated by higher-order Fibonacci numbers has been introduced and investigated from a spectral and norm-theoretic perspective. Explicit expressions for the eigenvalues of these matrices were derived by employing the Binet formula together with the structural properties of r-circulant matrices. Based on the obtained eigenvalue representations, a closed-form formula for the determinant was established.

Furthermore, several important matrix norms, including the Euclidean norm, the one-norm, the ∞-norm, and the spectral norm, were analyzed in detail. Lower and upper bounds for the spectral norm were obtained separately for cases $\left|r\right|\ge 1$ and $\left|r\right|<1$ by means of Hadamard product techniques and classical norm inequalities. The results reveal that the parameter r plays a decisive role in the spectral behavior of the matrices, leading to fundamentally different norm estimates in these two regimes.

Finally, numerical examples were presented to illustrate and validate the theoretical findings. The proposed framework not only extends several known results for circulant and r-circulant matrices with classical Fibonacci-type entries, but also provides a unified approach that can be applied to other families of special number sequences. Future research directions may include analogous investigations for r-circulant matrices associated with other generalized Fibonacci-type sequences, as well as potential applications in numerical analysis and signal processing.

While the present work is theoretical in nature, we note that structured matrices including circulant and r-circulant families continue to arise in a wide range of modern computational and engineering applications. For instance, explicit spectral and norm properties of structured matrices are relevant to large-scale graph processing [24], dynamic matrix inversion via neural networks [25], singular-value-based fault diagnosis in industrial systems [26], and structured tensor models for signal processing [27]. Although these works do not directly concern Fibonacci-type r-circulant matrices, they illustrate the broader significance of deriving explicit eigenvalue, determinant, and norm formulas for structured matrix classes, which is the primary contribution of the present paper.