On r-circulant matrices with generalized bi-periodic Fibonacci numbers

Abstract

In this paper, we calculate the Frobenius norm, and give upper and lower bounds for the spectral norm of r-circulant matrices whose entries are defined in terms of generalized bi-periodic Fibonacci numbers. We also provide explicit formulas for the computation of eigenvalues and determinants of these matrices.

1 Introduction

The generalized bi-periodic Fibonacci sequence $\{w_{n}\} = \{w_{n}(w_{0},w_{1};a,b)\}$, with arbitrary initial values $w_{0}$ and $w_{1}$, is defined [7] by the recurrence relation

$$\begin{aligned} w_{n}= & {} a^{\xi (n+1)}b^{\xi (n)}w_{n-1}+w_{n-2},\quad n \ge 2, \end{aligned}$$

where $\xi (n)= \left( 1-(-1)^{n} \right) /2$, and a and b are nonzero real numbers. Note that $\xi (n)$ returns to 0 when n is even, and to 1 when n is odd. Several well-known integer sequences are its special cases. For example, this sequence is reduced to the bi-periodic Fibonacci sequence $\{q_{n}\}$ for $w_{0}=0,w_{1}=1$, and to the bi-periodic Lucas sequence $\{p_{n}\}$ for $w_{0}=2,w_{1}=b$. We refer to [4, 7, 13, 18,19,20,21] for basic properties of these sequences and their generalizations.

Several recent works have been dedicated to the study of r-circulant matrices with special entries such as Fibonacci-like numbers. Solak [15, 16] obtained some bounds for the spectral norm of circulant matrices whose entries are Fibonacci and Lucas numbers. Shen and Cen [14] generalized the results of Solak to r-circulant matrices. Nalli and Sen [12] investigated the norms of circulant matrices with generalized Fibonacci numbers. Alptekin et al. [1] obtained the spectral norm and eigenvalues of circulant matrices whose entries are Horadam numbers. Yazlik and Taskara [23] found upper and lower bounds on the norms of r-circulant matrices with generalized k-Horadam numbers. They also provided formulas for the computation of the determinant and eigenvalues of such matrices. We refer to [2, 3, 5, 10, 11] for related studies.

Recently, Köme and Yazlik [9] obtained upper and lower bounds for the spectral norm of r-circulant matrices whose entries are bi-periodic Fibonacci and Lucas numbers. In the same spirit, we shall calculate the Frobenius norm, find upper and lower bounds on the spectral norm, and calculate the eigenvalues and determinants of r-circulant matrices whose entries are generalized bi-periodic Fibonacci numbers. To this purpose, we review the background material concerning the basic definitions and facts of r-circulant matrices and matrix norms in the rest of this section.

For $n>0$, the Binet formula of the sequence $\{w_{n}\}$ can be written as

$$\begin{aligned} w_{n}= & {} \frac{a^{\xi \left( n+1\right) }}{\left( ab\right) ^{\left\lfloor \frac{n}{2}\right\rfloor }}\left( X \alpha ^{n}-Y \beta ^{n}\right) , \end{aligned}$$

where

$$\begin{aligned} X =\frac{w_{1}-({\beta }/{a})w_{0}}{\alpha -\beta } \quad \text {and}\quad Y =\frac{w_{1}-({\alpha }/{a})w_{0}}{\alpha -\beta } \end{aligned}$$

[19, 20]. The numbers

$$\begin{aligned} \alpha =\frac{ab+\sqrt{a^{2}b^{2}+4ab}}{2} \quad \text {and}\quad \beta =\frac{ab-\sqrt{a^{2}b^{2}+4ab}}{2} \end{aligned}$$

(1)

are the roots of the polynomial $x^{2}-abx-ab$, and they satisfy

$$\begin{aligned} \alpha + \beta =ab,\quad \alpha -\beta =\sqrt{a^{2}b^{2}+4ab},\quad \alpha \beta =-ab. \end{aligned}$$

By the Binet formula, the sequences $\{q_{n}\}$ and $\{p_{n}\}$ are given by

$$\begin{aligned} q_{n} = \frac{a^{\xi \left( n+1\right) }}{\left( ab\right) ^{\left\lfloor \frac{n}{2}\right\rfloor }}\left( \frac{\alpha ^{n}-\beta ^{n}}{\alpha -\beta }\right) \quad \text {and}\quad p_{n}=\frac{a^{-\xi \left( n\right) }}{\left( ab\right) ^{\left\lfloor \frac{n}{2}\right\rfloor }}\left( \alpha ^{n}+\beta ^{n}\right) . \end{aligned}$$

(2)

Let $r\in \mathbb {C} \setminus \left\{ 0\right\}$. An $n\times n$ matrix $C_{r}=\begin{bmatrix} c_{ij} \end{bmatrix}$ with entries

$$\begin{aligned} c_{ij} = \left\{ \begin{array}{cl} c_{j-i}, &{} \ j\ge i ,\\ rc_{n+j-i}, &{} \ j<i, \end{array} \right. \end{aligned}$$

is called an r-circulant matrix. In other words, $C_{r}$ has the following form:

$$\begin{aligned} C_{r}= & {} \left[ \begin{array}{cccccc} c_{0} &{} c_{1} &{} c_{2} &{} \dots &{} c_{n-2} &{} c_{n-1} \\ rc_{n-1} &{} c_{0} &{} c_{1} &{} \dots &{} c_{n-3} &{} c_{n-2} \\ rc_{n-2} &{} rc_{n-1} &{} c_{0} &{} \dots &{} c_{n-4} &{} c_{n-3} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ rc_{2} &{} rc_{3} &{} rc_{4} &{} \dots &{} c_{0} &{} c_{1} \\ rc_{1} &{} rc_{2} &{} rc_{3} &{} \dots &{} rc_{n-1} &{} c_{0} \end{array} \right] . \end{aligned}$$

For simplicity, we denote $C_{r}$ by ${\text {circ}}_{r}\left[ c_{0},c_{1},\dots ,c_{n-1}\right]$. Note that $C_r$ is reduced to a circulant matrix for $r=1$. The eigenvalues of $C_{r}$ are given as

$$\begin{aligned} \lambda _{j}\left( C_{r}\right) =\sum _{k=0}^{n-1}c_{k}\left( \rho \omega ^{-j}\right) ^{k} \end{aligned}$$

(3)

with $j=0,1,\ldots ,n-1$, where $\rho$ is any nth root of r, and $\omega$ is any nth root of unity. For details, we refer to [6, Lemma 4]. An eigenvalue formula was provided in [23, Theorem 7] for r-circulant matrices with k-Horadam numbers. With suitable initial values and polynomials, this formula contains the eigenvalues of r-circulants with Fibonacci numbers and several other Fibonacci-like numbers as special cases.

Let $A= \begin{bmatrix} a_{ij} \end{bmatrix}$ be an $m\times n$ matrix. The Frobenius norm (also known as Hilbert-Schmidt norm or Schur norm) $\left\| A\right\| _{F}$ of A is the square root of the sum of the squares of the absolute values of all entries of A. That is,

$$\begin{aligned} \left\| A\right\| _{F}= & {} \sqrt{\sum _{i=1}^{m}\sum _{j=1}^{n}\left| a_{ij}\right| ^{2}}. \end{aligned}$$

Another important norm of A is the spectral norm, defined as

$$\begin{aligned} \left\| A\right\| _{2}= & {} \sqrt{\lambda _{\max }\left( A^{*}A\right) }, \end{aligned}$$

where $\lambda _{\max }\left( A^{*} A\right)$ denotes the largest eigenvalue of $A^{*}A$. Here, $A^{*}$ is the conjugate transpose of A. The following inequality by Stone [17] provides a relationship between Frobenius and spectral norms:

$$\begin{aligned} \begin{array}{ccccccc} \frac{1}{\sqrt{n}}\left\| A\right\| _{F}\le & {} \left\| A\right\| _{2}\le & {} \left\| A\right\| _{F}\le & {} \sqrt{n}\left\| A\right\| _{2}. \end{array} \end{aligned}$$

(4)

Note that the Frobenius norm is an upper bound on the spectral norm.

For $m\times n$ matrices $A=\begin{bmatrix} a_{ij} \end{bmatrix}$ and $B=\begin{bmatrix} b_{ij} \end{bmatrix}$, the Hadamard product of A and B is defined as $A\circ B = \begin{bmatrix} a_{ij} \cdot b_{ij}\end{bmatrix}$. It is simply the entrywise multiplication of A and B. This product appears [8, Theorem 5.5.3] in

$$\begin{aligned} \begin{array}{ccc} \left\| A\circ B \right\| _{2}\le & {} r_{1}(A) c_{1}(B). \end{array} \end{aligned}$$

Here,

$$\begin{aligned} \begin{array}{ccccccccc} r_{1}(A)= & {} \displaystyle \max _{1\le i\le m}\sqrt{\sum _{j=1}^{n} \left| a_{ij}\right| ^{2}}&\,&\text { and }&\,&c_{1}( B)= & {} \displaystyle \max _{1\le j\le n}\sqrt{\sum _{i=1}^{m} \left| b_{ij}\right| ^{2}} \end{array}. \end{aligned}$$

Note that $r_{1}(A)$ is the maximum row length norm of A, and $c_{1}(B)$ is the maximum column length norm of B.

2 Main results

Throughout this section, we let a, b and $w_1$ be positive integers and let $w_0$ be a nonnegative integer unless otherwise is stated.

We study the matrix

$$\begin{aligned} W_{r}= & {} {\text {circ}}_{r}\left[ \left( \frac{a}{b}\right) ^{\frac{\xi (0) }{2}} w_{0},\, \left( \frac{a}{b}\right) ^{\frac{\xi (1) }{2}} w_{1}\,,\ldots ,\, \left( \frac{a}{b}\right) ^{\frac{\xi (n-1) }{2}} w_{n-1} \right] . \end{aligned}$$

Lemma 1

For $n>1$, we have

$$\begin{aligned} \sum _{k=1}^{n}\left( \frac{a}{b}\right) ^{\xi \left( k\right) }w_{k}^{2}= & {} \frac{1}{b}\left( w_{n}w_{n+1}-w_{0}w_{1}\right) . \end{aligned}$$

Proof

Recall that the Binet formula of the sequence $\{w_{k}\}$ is given by

$$\begin{aligned} w_{k}= & {} \frac{a^{\xi (k+1)}}{(ab)^{\left\lfloor \frac{k}{2} \right\rfloor }} \left( X \alpha ^{k}-Y \beta ^{k}\right) . \end{aligned}$$

Since $\xi (n)+\xi (n+1)=1$ and $\lfloor n/2\rfloor +\lfloor (n+1)/2 \rfloor =n$, we have

$$\begin{aligned} w_{n}w_{n+1}= & {} \frac{a^{\xi (n+1)+\xi (n) }}{(ab)^{\left\lfloor \frac{n}{2}\right\rfloor + \left\lfloor \frac{n+1}{2}\right\rfloor }} \left( X \alpha ^{n} - Y \beta ^{n}\right) \left( X \alpha ^{n+1} - Y \beta ^{n+1} \right) \nonumber \\= & {} \frac{a}{(ab)^{n}}\left[ X ^{2} \alpha ^{2n+1} - XY (\alpha \beta )^{n} ( \alpha + \beta ) +Y ^{2} \beta ^{2n+1}\right] \nonumber \\= & {} \frac{a}{(ab)^{n}}\left[ X ^{2} \alpha ^{2n+1} + Y ^{2} \beta ^{2n+1} - XY (-ab)^{n} (ab) \right] . \end{aligned}$$

(5)

On the other hand, we have

$$\begin{aligned} w_{k}^{2}= & {} \frac{a^{2\xi (k+1) }}{(ab)^{2\left\lfloor \frac{k}{2}\right\rfloor }}\left[ X \alpha ^{k} - Y \beta ^{k}\right] ^{2} \\= & {} \frac{a^{2\xi (k+1) }}{(ab)^{2\left\lfloor \frac{k}{2}\right\rfloor }}\left[ X ^{2} \alpha ^{2k} + Y ^{2} \beta ^{2k} - 2XY ( \alpha \beta )^{k} \right] . \end{aligned}$$

Now, if k is even, we get

$$\begin{aligned} w_{k}^{2}= & {} \frac{a^{2}}{(ab)^{k}} \left[ X ^{2} \alpha ^{2k} + Y ^{2} \beta ^{2k} - 2XY (\alpha \beta )^{k} \right] , \end{aligned}$$

and if k is odd,

$$\begin{aligned} w_{k}^{2}= & {} \frac{ab}{(ab)^{k}} \left[ X ^{2} \alpha ^{2k} + Y ^{2} \beta ^{2k} - 2XY (\alpha \beta )^{k} \right] . \end{aligned}$$

Since $\alpha \beta =-ab$ and

$$\begin{aligned} a^{2}\left( \frac{b}{a}\right) ^{\xi (k)}= & {} \left\{ \begin{array}{cl} a^{2}, &{}\ \text {if } k \text { is even}, \\ ab, &{}\ \text {if } k \text { is odd}, \end{array} \right. \end{aligned}$$

we can write

$$\begin{aligned} w_{k}^{2}= & {} a^{2} \left( \frac{b}{a} \right) ^{\xi (k)} \left[ X ^{2} \left( \frac{\alpha ^{2}}{ab}\right) ^{k} + Y ^{2}\left( \frac{\beta ^{2}}{ab}\right) ^{k} - 2XY (-1)^{k} \right] , \end{aligned}$$

or equivalently,

$$\begin{aligned} a^{-2} \left( \frac{a}{b} \right) ^{\xi (k) }w_{k}^{2}= & {} X ^{2} \left( \frac{\alpha ^{2}}{ab}\right) ^{k} + Y ^{2} \left( \frac{\beta ^{2}}{ab}\right) ^{k}-2XY (-1)^{k}. \end{aligned}$$

(6)

By using the geometric sum formula, it can be seen that

$$\begin{aligned} \sum _{k=1}^{n}\left( \frac{\alpha ^{2}}{ab}\right) ^{k} = \frac{\alpha ^{2n+1}}{(ab)^{n+1}}-\frac{\alpha }{ab} \quad \text {and}\quad \sum _{k=1}^{n}\left( \frac{\beta ^{2}}{ab}\right) ^{k} = \frac{\beta ^{2n+1}}{(ab)^{n+1}}-\frac{\beta }{ab}. \end{aligned}$$

Now we take the summation of both sides of Eq. (6) from 1 to n:

$$\begin{aligned} \begin{aligned}&\sum _{k=1}^{n}a^{-2}\left( \frac{a}{b}\right) ^{\xi \left( k\right) }w_{k}^{2} \ =\ X ^{2} \sum _{k=1}^{n} \left( \frac{\alpha ^{2}}{ab}\right) ^{k} + Y ^{2} \sum _{k=1}^{n} \left( \frac{\beta ^{2}}{ab}\right) ^{k} - XY \sum _{k=1}^{n} 2(-1)^{k} \\&\quad = X ^{2}\left[ \frac{\alpha ^{2n+1}}{(ab)^{n+1}} - \frac{\alpha }{ab} \right] + Y ^{2} \left[ \frac{\beta ^{2n+1}}{(ab)^{n+1}} - \frac{\beta }{ab}\right] - XY \left[ ( -1)^{n}-1 \right] \\&\quad = \frac{1}{ab} \left[ X ^{2}\alpha ^{2n+1} \left( \frac{1}{ab}\right) ^{\! n} \!\!+ Y ^{2} \beta ^{2n+1} \left( \frac{1}{ab}\right) ^{\! n}\!\! - X ^{2}\alpha - Y ^{2} \beta - XY ab \left[ \left( -1\right) ^{n}\! -1 \right] \! \right] \! . \end{aligned} \end{aligned}$$

By taking Equation (5) into account in the last line of the equation above, we get the desired result:

$$\begin{aligned} \sum _{k=1}^{n}a^{-2}\left( \frac{a}{b}\right) ^{\xi \left( k\right) }w_{k}^{2}= & {} \frac{1}{ab}\frac{1}{a} ( w_{n}w_{n+1} ) + XY - \frac{X ^{2} \alpha }{ab} - \frac{Y ^{2}\beta }{ab} \\= & {} \frac{1}{ab} \left[ \frac{1}{a} ( w_{n}w_{n+1} ) +XY ( ab) -\left( X ^{2}\alpha +Y ^{2}\beta \right) \right] \\= & {} \frac{1}{ab} \left[ \frac{1}{a} w_{n}w_{n+1} - \frac{1}{a} w_{0}w_{1} \right] \\= & {} \frac{1}{a^{2}b} \big [ w_{n}w_{n+1} -w_{0}w_{1} \big ] . \end{aligned}$$

$\square$

An immediate consequence of Lemma 1 is the following.

Corollary 1

For $n>0$,

$$\begin{aligned} \sum _{k=0}^{n-1} \left( \frac{a}{b}\right) ^{\xi (k)} w_{k}^{2}= & {} \frac{1}{b} \left( w_{n}w_{n-1}-w_{0}w_{1}+bw_{0}^{2} \right) . \end{aligned}$$

Remark 1

If we take the initial values $w_{0}=0$ and $w_{1}=1$, we get

$$\begin{aligned} \sum _{k=1}^{n}\left( \frac{a}{b}\right) ^{\xi \left( k\right) }q_{k}^{2}= & {} \frac{1}{b}q_{n}q_{n+1}. \end{aligned}$$

This identity was given in [22, Theorem 2.3]. Similarly, with the initial values $w_{0}=2$ and $w_{1}=b$, we get [9, Theorem 2.1]:

$$\begin{aligned} \sum _{k=1}^{n}\left( \frac{a}{b}\right) ^{\xi \left( k\right) }p_{k}^{2}= & {} \frac{1}{b}p_{n}p_{n+1}-2. \end{aligned}$$

Now we are ready to provide bounds for $\left\| W_{r}\right\| _{2}$. But let us first calculate $\left\| W_{r}\right\| _{F}$.

Lemma 2

The Frobenius norm

$$\begin{aligned} \left\| W_{r} \right\| _{F}= & {} \sqrt{\sum _{k=0}^{n-1} \left( n+k\left( \left| r\right| ^{2}-1\right) \right) \left( \frac{a}{b}\right) ^{\xi \left( k\right) }w_{k}^{2} }. \end{aligned}$$

Proof

By using Lemma 1 and Corollary 1, it is clear that

$$\begin{aligned} \left\| W_{r}\right\| _{F}= & {} \sqrt{ \sum _{k=0}^{n-1} ( n-k) \left( \frac{a}{b}\right) ^{\xi (k)} w_{k}^{2} + \sum _{k=1}^{n-1}k \left| r \right| ^{2} \left( \frac{a}{b}\right) ^{\xi (k) }w_{k}^{2}} \\= & {} \sqrt{\sum _{k=0}^{n-1} \left( n+k \left( \left| r \right| ^{2} - 1\right) \right) \left( \frac{a}{b}\right) ^{\xi (k) }w_{k}^{2}}. \end{aligned}$$

$\square$

Theorem 1

Let

$$\begin{aligned} \varDelta = w_{n-1}w_{n}-w_{0}w_{1}+b w_{0}^{2}. \end{aligned}$$

• (i) If $|r|\ge 1$, then

  $$\begin{aligned} \begin{array}{rlllll} \sqrt{\frac{\varDelta }{b}}\le & {} \left\| W_{r}\right\| _{2}\le & {} \sqrt{\left( (n-1) \left| r\right| ^{2}+1 \right) \frac{\varDelta }{b}}. \end{array} \end{aligned}$$

• (ii) If $|r|<1$, then

  $$\begin{aligned} \begin{array}{rlllll} \left| r \right| \sqrt{\frac{\varDelta }{b}}\le & {} \left\| W_{r}\right\| _{2}\le & {} \sqrt{n\frac{\varDelta }{b}}. \end{array} \end{aligned}$$

Proof

(i) Let $\left| r\right| \ge 1$. From Corollary 1 and Lemma 2, we have

$$\begin{aligned} \begin{array}{rlllll} \left\| W_{r} \right\| _{F}\ge & {} \sqrt{ \displaystyle \sum _{k=0}^{n-1}n \left( \frac{a}{b}\right) ^{\xi (k) }w_{k}^{2}}= & {} \sqrt{n\frac{\varDelta }{b}}. \end{array} \end{aligned}$$

Therefore, we can write

$$\begin{aligned} \frac{1}{\sqrt{n}}\left\| W_{r}\right\| _{F}\ge & {} \sqrt{\frac{\varDelta }{b}}. \end{aligned}$$

From (4), we obtain

$$\begin{aligned} \sqrt{\frac{\varDelta }{b}}\le & {} \left\| W_{r}\right\| _{2}. \end{aligned}$$

In order to provide an upper bound, let

$$\begin{aligned} U = {\text {circ}}_{r}\left[ 1,1,\dots ,1 \right] \quad \text {and}\quad W = W_1. \end{aligned}$$

Then

$$\begin{aligned} W_r = U \circ W. \end{aligned}$$

Since $|r|\ge 1$, we have

$$\begin{aligned} \begin{array}{ccccccc} r_{1}\left( U\right)= & {} \displaystyle \max _{1\le i\le n}\sqrt{\sum _{j=1}^{n}\left| u_{ij}\right| ^{2}}= & {} \displaystyle \sqrt{\sum _{j=1}^{n}\left| u_{nj}\right| ^{2}}= & {} \displaystyle \sqrt{\left( n-1\right) \left| r\right| ^{2}+1} \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ccccccc} c_{1}\left( W\right)= & {} \displaystyle \max _{1\le j\le n}\sqrt{\sum _{i=1}^{n} \left| w_{ij}\right| ^{2}}= & {} \displaystyle \sqrt{\sum _{k=0}^{n-1}\left( \frac{a}{b}\right) ^{\xi \left( k\right) } w_{k}^{2}}= & {} \displaystyle \sqrt{\frac{\varDelta }{b}}. \end{array} \end{aligned}$$

Using the above quantities, we obtain

$$\begin{aligned} \begin{array}{ccccccc} \left\| W_{r}\right\| _{2}= & {} \left\| U\circ W\right\| _{2}\le & {} r_{1}\left( U\right) c_{1}\left( W\right)= & {} \sqrt{\left( (n-1)|r|^{2}+1\right) \frac{\varDelta }{b}}. \end{array} \end{aligned}$$

(ii) Let $|r|<1.$ Suppose k is an integer with $0\le k\le n-1$. Since $|r|^{2}-1<0$, the minimum of $n+k(|r|^{2}-1)$ is achieved when $k=n-1$. So, for $k=n-1$ we have $n+k(|r|^{2}-1)=n|r|^{2}-|r|^{2}+1\ge n|r|^{2}$. Then

$$\begin{aligned} n+k(|r|^{2}-1)\ge & {} n|r|^{2} \end{aligned}$$

for each k with $0\le k\le n-1$. Therefore, we can write

$$\begin{aligned} \left\| W_{r} \right\| _{F}= & {} \sqrt{\sum _{k=0}^{n-1} \left( n+k \left( \left| r \right| ^{2} - 1\right) \right) \left( \frac{a}{b} \right) ^{\xi (k) }w_{k}^{2}} \\\ge & {} \sqrt{\sum _{k=0}^{n-1} n \left| r\right| ^{2}\left( \frac{a}{b}\right) ^{\xi (k) }w_{k}^{2}}. \end{aligned}$$

Then it follows that

$$\begin{aligned} \frac{1}{\sqrt{n}} \left\| W_{r} \right\| _{F}\ge & {} \left| r \right| \sqrt{\frac{\varDelta }{b}}. \end{aligned}$$

By (4), we get

$$\begin{aligned} \left\| W_{r} \right\| _{2}\ge & {} \left| r \right| \sqrt{\frac{\varDelta }{b}}. \end{aligned}$$

In order to provide an upper bound, we have

$$\begin{aligned} \begin{array}{ccccccc} r_{1} (U)= & {} \displaystyle \max _{1\le i\le n}\sqrt{\sum _{j=1}^{n} \left| u_{ij}\right| ^{2}}= & {} \displaystyle \sqrt{\sum _{j=1}^{n} \left| u_{1j} \right| ^{2}}= & {} \displaystyle \sqrt{n} \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ccccccc} c_{1} (W)= & {} \displaystyle \max _{1\le j\le n}\sqrt{\sum _{i=1}^{n}\left| w_{ij}\right| ^{2}}= & {} \displaystyle \sqrt{\sum _{k=0}^{n-1}\left( \frac{a}{b}\right) ^{\xi (k) }w_{k}^{2}}= & {} \displaystyle \sqrt{\frac{\varDelta }{b}}. \end{array} \end{aligned}$$

In conclusion,

$$\begin{aligned} \begin{array}{ccccccc} \left\| W_{r} \right\| _{2}= & {} \left\| U\circ W\right\| _{2}\le & {} r_{1} (U) c_{1} (W)= & {} \sqrt{n\frac{\varDelta }{b}}. \end{array} \end{aligned}$$

$\square$

Remark 2

We can use Theorem 1 to provide bounds for special cases.

(i):

If $w_{0}=0$ and $w_{1}=1$, then

$$\begin{aligned} \begin{array}{rlllll} \sqrt{\frac{q_{n-1}q_{n}}{b}} &{}\le &{} \left\| W_{r}\right\| _{2} &{}\le &{} \sqrt{\big ( \left( n-1\right) \left| r\right| ^{2}+1\big ) \frac{q_{n-1}q_{n}}{b}},&{} \quad |r|\ge 1, \\ \left| r\right| \sqrt{\frac{q_{n-1}q_{n}}{b}} &{}\le &{} \left\| W_{r}\right\| _{2} &{}\le &{} \sqrt{n\frac{q_{n-1}q_{n}}{b}}, &{} \quad |r|<1. \end{array} \end{aligned}$$

(ii):

If $w_{0}=2$ and $w_{1}=b$, then

$$\begin{aligned} \begin{array}{rlllll} \sqrt{\frac{p_{n-1}p_{n}}{b}+2} &{}\le &{} \left\| W_{r}\right\| _{2} &{}\le &{} \sqrt{\big ( \left( n-1\right) \left| r\right| ^{2}+1\big ) \big ( \frac{p_{n-1}p_{n}}{b}+2 \big ) }, &{} \quad |r|\ge 1, \\ \left| r\right| \sqrt{\frac{p_{n-1}p_{n}}{b}+2} &{}\le &{} \left\| W_{r}\right\| _{2} &{}\le &{} \sqrt{n \big ( \frac{p_{n-1}p_{n}}{b}+2 \big ) },&{} \quad |r|<1. \end{array} \end{aligned}$$

(iii):

Finally, if $a = b =1$, then

$$\begin{aligned} \begin{array}{rlllll} \sqrt{\varDelta } &{}\le &{} \left\| W_{r}\right\| _{2} &{}\le &{} \sqrt{\big ( \left( n-1\right) \left| r\right| ^{2}+1\big ) \varDelta },&{} \quad |r|\ge 1, \\ \left| r\right| \sqrt{\varDelta } &{}\le &{} \left\| W_{r}\right\| _{2} &{}\le &{} \sqrt{n \varDelta }, &{}\quad |r|<1. \end{array} \end{aligned}$$

Note that $\varDelta = w_{n-1}w_{n}-w_{0}w_{1} + w_{0}^{2}$ since $b=1$.

Remark 3

The lower bounds in the first two parts of Remark 2 equal those of [9, Theorems 2.2–3]. However, the upper bounds are weaker except for the case that $|r|<1$ in the second part. The weakness is caused by the choice of the matrices U and W in the Hadamard product. Special initial values in [9] allow a flexibility to make better choices for U and W in order to improve the bounds. So, Theorem 1 extends [9, Theorems 2.2–3] as to the lower bounds and one upper, but not as to the remaining upper bounds. The bounds in Remark 2(iii) can be found in [5].

Theorem 2

The eigenvalues

$$\begin{aligned} \lambda _{j} (W_{r}) = \frac{\left( \frac{a}{b}\right) ^{\frac{\xi (n)}{2}} rw_{n} - w_{0} + \rho \omega ^{-j} \left[ \left( \frac{a}{b}\right) ^{{\frac{\xi \left( n+1\right) }{2}}}rw_{n-1}+\left( \frac{a}{b}\right) ^{^{\frac{1}{2}}}\left( bw_{0}-w_{1}\right) \right] }{\rho ^{2} \omega ^{-2j} + (ab)^{\frac{1}{2}}\rho \omega ^{-j}-1}, \end{aligned}$$

for $j=0,1,\dots ,n-1$, provided that

$$\begin{aligned} r \ne \left( \frac{-\alpha }{\beta } \right) ^{\pm \frac{n}{2}}. \end{aligned}$$

Here, $\rho = r^{\frac{1}{n}}$ and $\omega$ is any nth root of unity. For $\alpha$ and $\beta$, see (1).

Proof

From (3), we have

$$\begin{aligned} \lambda _{j} (W_{r})= & {} \sum _{k=0}^{n-1} \left( \frac{a}{b}\right) ^{\frac{\xi (k)}{2}} w_{k} \rho ^{k} \omega ^{-kj} \nonumber \\= & {} X a \sum _{k=0}^{n-1} \left( \frac{\alpha \rho \omega ^{-j}}{(ab)^{\frac{1}{2}}} \right) ^{k} - Y a \sum _{k=0}^{n-1}\left( \frac{\beta \rho \omega ^{-j}}{(ab)^{\frac{1}{2}}} \right) ^{k} \\= & {} \frac{a}{(ab)^{\frac{n-1}{2}}} \left( X \frac{\alpha ^{n} r - (ab) ^{\frac{n}{2}}}{\alpha \rho \omega ^{-j} - (ab)^{\frac{1}{2}}} - Y \frac{\beta ^{n} r - ( ab)^{\frac{n}{2}}}{\beta \rho \omega ^{-j} - (ab)^{\frac{1}{2}}}\right) \nonumber \\= & {} \frac{a}{(ab)^{\frac{n-1}{2}} \left( \alpha \rho \omega ^{-j} - (ab)^{\frac{1}{2}} \right) \left( \beta \rho \omega ^{-j} - (ab)^{\frac{1}{2}} \right) } \nonumber \\&\times \left[ r \rho \omega ^{-j} \alpha \beta \left( X \alpha ^{n-1} - Y \beta ^{n-1}\right) - r (ab)^{\frac{1}{2}} \left( X \alpha ^{n}-Y \beta ^{n} \right) \right. \nonumber \\&\left. -\rho \omega ^{-j} (ab)^{\frac{n}{2}} \left( X \beta - Y \alpha \right) + (ab)^{\frac{n+1}{2}} \left( X -Y \right) \right] . \nonumber \end{aligned}$$

(7)

Therefore, after lengthy calculations,

$$\begin{aligned} \lambda _{j} (W_{r})= & {} \frac{rw_{n}-w_{0} + \left( \frac{a}{b}\right) ^{\frac{1}{2}} \rho \omega ^{-j} \Big [ rw_{n-1}+\left( bw_{0}-w_{1}\right) \Big ] }{\rho ^{2} \omega ^{-2j} + (ab)^{\frac{1}{2}} \rho \omega ^{-j} - 1}, \end{aligned}$$

when n is even, and

$$\begin{aligned} \lambda _{j} (W_{r})= & {} \frac{\left( \frac{a}{b}\right) ^{\frac{1}{2}} rw_{n} - w_{0} + \rho \omega ^{-j} \Big [ rw_{n-1} + \left( \frac{a}{b}\right) ^{\frac{1}{2}}(bw_{0}-w_{1}) \Big ] }{\rho ^{2}\omega ^{-2j}+ (ab)^{\frac{1}{2}} \rho \omega ^{-j} - 1}, \end{aligned}$$

when n is odd. This completes the proof. $\square$

Remark 4

The geometric sum formula fails in (7) when $\alpha \rho \omega ^{-j}/ (ab)^{\frac{1}{2}} =1$ or$\beta \rho \omega ^{-j} / (ab)^{\frac{1}{2}}=1$. Therefore, the assumption $r \ne (-\alpha / \beta )^{\pm \frac{n}{2}}$ is needed for it to be valid. This also guarantees that the denominator $\rho ^{2} \omega ^{-2j} + (ab)^{\frac{1}{2}}\rho \omega ^{-j}-1 \ne 0$.

Theorem 3

The determinant

$$\begin{aligned} \det ( W_{r}) = \frac{\left[ w_{0} - \left( \frac{a}{b} \right) ^{\frac{\xi (n)}{2}} r w_{n} \right] ^{n} - r \left[ \left( \frac{a}{b}\right) ^{\frac{\xi (n+1) }{2}} r w_{n-1} + \left( \frac{a}{b}\right) ^{\frac{1}{2}} ( bw_{0}-w_{1}) \right] ^{n}}{1-\left( \frac{a}{b}\right) ^{\frac{\xi (n) }{2}}p_{n} r+(-1)^{n}\, r^{2}}, \end{aligned}$$

provided that

$$\begin{aligned} r \ne \left( \frac{-\alpha }{\beta } \right) ^{\pm \frac{n}{2}}. \end{aligned}$$

Here, $p_n$ is as in (2).

Proof

The formula follows from the fact that $\det \left( W_{r}\right) =\displaystyle \prod \limits _{j=0}^{n-1}\lambda _{j}\left( W_{r}\right)$. $\square$

3 Conclusion

In this paper, we obtained bounds on the spectral norm of r-circulant matrices whose entries are generalized bi-periodic Fibonacci numbers. We also calculated the eigenvalues and determinants of these matrices explicitly. By means of this work, we have a unified approach for dealing with many r-circulant matrices with special entries such as Fibonacci, Lucas, Pell, Pell-Lucas, generalized Fibonacci, bi-periodic Fibonacci, and bi-periodic Lucas numbers. We note that our bounds can be improved for specified initial values by choosing suitable matrices in the Hadamard product.