Open Access

Some algebraic identities on quadra Fibona-Pell integer sequence

Advances in Difference Equations20152015:148

https://doi.org/10.1186/s13662-015-0486-7

Received: 25 March 2015

Accepted: 26 April 2015

Published: 8 May 2015

Abstract

In this work, we define a quadra Fibona-Pell integer sequence \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\) for \(n\geq4\) with initial values \(W_{0}=W_{1}=0\), \(W_{2}=1\), \(W_{3}=3\), and we derive some algebraic identities on it including its relationship with Fibonacci and Pell numbers.

Keywords

Fibonacci numbers Lucas numbers Pell numbers Binet’s formula binary linear recurrences

1 Preliminaries

Let p and q be non-zero integers such that \(D=p^{2}-4q\neq0\) (to exclude a degenerate case). We set the sequences \(U_{n}\) and \(V_{n}\) to be
$$ \begin{aligned} &U_{n}=U_{n}(p,q)=pU_{n-1}-qU_{n-2}, \\ &V_{n}=V_{n}(p,q)=pV_{n-1}-qV_{n-2} \end{aligned} $$
(1)
for \(n\geq2\) with initial values \(U_{0}=0\), \(U_{1}=1\), \(V_{0}=2\), and \(V_{1}=p\). The sequences \(U_{n}\) and \(V_{n}\) are called the (first and second) Lucas sequences with parameters p and q. \(V_{n}\) is also called the companion Lucas sequence with parameters p and q.
The characteristic equation of \(U_{n}\) and \(V_{n}\) is \(x^{2}-px+q=0\) and hence the roots of it are \(x_{1}=\frac{p+\sqrt{D}}{2}\) and \(x_{2}=\frac{p-\sqrt{D}}{2}\). So their Binet formulas are
$$U_{n}=\frac{x_{1}^{n}-x_{2}^{n}}{x_{1}-x_{2}} \quad \mbox{and}\quad V_{n}=x_{1}^{n}+x_{2}^{n} $$
for \(n\geq0\). For the companion matrix \(M=\bigl[ {\scriptsize\begin{matrix} p & -q \cr 1 & 0\end{matrix}} \bigr] \), one has
$$ \left [ \begin{array}{@{}c@{}} U_{n} \\ U_{n-1} \end{array} \right ] =M^{n-1}\left [ \begin{array}{@{}c@{}} 1 \\ 0 \end{array} \right ]\quad \mbox{and} \quad \left [ \begin{array}{@{}c@{}} V_{n} \\ V_{n-1}\end{array} \right ] =M^{n-1}\left [ \begin{array}{@{}c@{}} p \\ 2\end{array} \right ] $$
for \(n\geq1\). The generating functions of \(U_{n}\) and \(V_{n}\) are
$$ U(x)=\frac{x}{1-px+qx^{2}}\quad \mbox{and} \quad V(x)=\frac{2-px}{1-px+qx^{2}}. $$
(2)

Fibonacci, Lucas, Pell, and Pell-Lucas numbers can be derived from (1). Indeed for \(p=1\) and \(q=-1\), the numbers \(U_{n}=U_{n}(1,-1)\) are called the Fibonacci numbers (A000045 in OEIS), while the numbers \(V_{n}=V_{n}(1,-1)\) are called the Lucas numbers (A000032 in OEIS). Similarly, for \(p=2\) and \(q=-1\), the numbers \(U_{n}=U_{n}(2,-1)\) are called the Pell numbers (A000129 in OEIS), while the numbers \(V_{n}=V_{n}(2,-1)\) are called the Pell-Lucas (A002203 in OEIS) (companion Pell) numbers (for further details see [16]).

2 Quadra Fibona-Pell sequence

In [7], the author considered the quadra Pell numbers \(D(n)\), which are the numbers of the form \(D(n)=D(n-2)+2D(n-3)+D(n-4)\) for \(n\geq4\) with initial values \(D(0)=D(1)=D(2)=1\), \(D(3)=2\), and the author derived some algebraic relations on it.

In [8], the authors considered the integer sequence (with four parameters) \(T_{n}=-5T_{n-1}-5T_{n-2}+2T_{n-3}+2T_{n-4}\) with initial values \(T_{0}=0\), \(T_{1}=0\), \(T_{2}=-3\), \(T_{3}=12\), and they derived some algebraic relations on it.

In the present paper, we want to define a similar sequence related to Fibonacci and Pell numbers and derive some algebraic relations on it. For this reason, we set the integer sequence \(W_{n}\) to be
$$ W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4} $$
(3)
for \(n\geq4\) with initial values \(W_{0}=W_{1}=0\), \(W_{2}=1\), \(W_{3}=3\) and call it a quadra Fibona-Pell sequence. Here one may wonder why we choose this equation and call it a quadra Fibona-Pell sequence. Let us explain: We will see below that the roots of the characteristic equation of \(W_{n}\) are the roots of the characteristic equations of both Fibonacci and Pell sequences. Indeed, the characteristic equation of (3) is \(x^{4}-3x^{3}+3x+1=0\) and hence the roots of it are
$$ \alpha=\frac{1+\sqrt{5}}{2},\qquad \beta=\frac{1-\sqrt{5}}{2},\qquad \gamma =1+ \sqrt{2} \quad \mbox{and}\quad \delta=1-\sqrt{2}. $$
(4)
(Here α, β are the roots of the characteristic equation of Fibonacci numbers and γ, δ are the roots of the characteristic equation of Pell numbers.) Then we can give the following results for \(W_{n}\).

Theorem 1

The generating function for \(W_{n}\) is
$$ W(x)=\frac{x^{2}}{x^{4}+3x^{3}-3x+1}. $$

Proof

The generating function \(W(x)\) is a function whose formal power series expansion at \(x=0\) has the form
$$ W(x)=\sum_{n=0}^{\infty}W_{n}x^{n}=W_{0}+W_{1}x+W_{2}x^{2}+ \cdots +W_{n}x^{n}+\cdots. $$
Since the characteristic equation of (3) is \(x^{4}-3x^{3}+3x+1=0\), we get
$$\begin{aligned} \bigl(1-3x+3x^{3}+x^{4}\bigr)W(x) =&\bigl(1-3x+3x^{3}+x^{4} \bigr) \bigl(W_{0}+W_{1}x+\cdots +W_{n}x^{n}+ \cdots\bigr) \\ =&W_{0}+(W_{1}-3W_{0})x+(W_{2}-3W_{1})x^{2} \\ &{}+(W_{3}-3W_{2}+3W_{0})x^{3}+\cdots \\ &{}+(W_{n}-3W_{n-1}+3W_{n-3}+W_{n-4})x^{n}+ \cdots. \end{aligned}$$
Notice that \(W_{0}=W_{1}=0\), \(W_{2}=1\), \(W_{3}=3\), and \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\). So \((1-3x+3x^{3}+x^{4})W(x)=x^{2}\) and hence the result is obvious. □

Theorem 2

The Binet formula for \(W_{n}\) is
$$ W_{n}= \biggl( \frac{\gamma^{n}-\delta^{n}}{\gamma-\delta } \biggr) - \biggl( \frac{\alpha^{n}-\beta^{n}}{\alpha-\beta } \biggr) $$
for \(n\geq0\).

Proof

Note that the generating function is \(W(x)=\frac{x^{2}}{x^{4}+3x^{3}-3x+1}\). It is easily seen that \(x^{4}+3x^{3}-3x+1=(1-x-x^{2})(1-2x-x^{2})\). So we can rewrite \(W(x)\) as
$$ W(x)=\frac{x}{1-2x-x^{2}}-\frac{x}{1-x-x^{2}}. $$
(5)
From (2), we see that the generating function for Pell numbers is
$$ P(x)=\frac{x}{1-2x-x^{2}} $$
(6)
and the generating function for the Fibonacci numbers is
$$ F(x)=\frac{x}{1-x-x^{2}}. $$
(7)
From (5), (6), (7), we get \(W(x)=P(x)-F(x)\). So \(W_{n}= ( \frac{\gamma^{n}-\delta^{n}}{\gamma-\delta } ) - ( \frac{\alpha^{n}-\beta^{n}}{\alpha-\beta } )\) as we wanted. □

The relationship with Fibonacci, Lucas, and Pell numbers is given below.

Theorem 3

For the sequences \(W_{n}\), \(F_{n}\), \(L_{n}\), and \(P_{n}\), we have:
  1. (1)

    \(W_{n}=P_{n}-F_{n}\) for \(n\geq0\).

     
  2. (2)

    \(W_{n+1}+W_{n-1}=(\gamma^{n}+\delta^{n})-(\alpha^{n}+\beta^{n})\) for \(n\geq1\).

     
  3. (3)

    \(\sqrt{5}F_{n}+2\sqrt{2}P_{n}=(\gamma^{n}-\delta^{n})+(\alpha ^{n}-\beta^{n})\) for \(n\geq1\).

     
  4. (4)

    \(L_{n}+P_{n+1}+P_{n-1}=\alpha^{n}+\beta^{n}+\gamma^{n}+\delta^{n}\) for \(n\geq1\).

     
  5. (5)

    \(2(W_{n+1}-W_{n}+F_{n-1})=\gamma^{n}+\delta^{n}\) for \(n\geq1\).

     
  6. (6)

    \(\lim_{n\rightarrow\infty}\frac{W_{n}}{W_{n-1}}=\gamma\).

     

Proof

(1) It is clear from the above theorem, since \(W(x)=P(x)-F(x)\).

(2) Since \(6W_{n-1}+W_{n+2}=3W_{n+1}-3W_{n-1}-W_{n-2}+6W_{n-1}\), we get
$$\begin{aligned} W_{n+1}+W_{n-1} =&2W_{n-1}+\frac{1}{3}W_{n-2}+ \frac{1}{3}W_{n+2} \\ =&\frac{6}{3} \biggl( \frac{\gamma^{n-1}-\delta^{n-1}}{\gamma-\delta }-\frac{\alpha^{n-1}-\beta^{n-1}}{\alpha-\beta} \biggr) \\ &{}+\frac{1}{3} \biggl( \frac{\gamma^{n-2}-\delta^{n-2}}{\gamma-\delta }-\frac{\alpha^{n-2}-\beta^{n-2}}{\alpha-\beta} \biggr) \\ &{}+\frac{1}{3} \biggl( \frac{\gamma^{n+2}-\delta^{n+2}}{\gamma-\delta }-\frac{\alpha^{n+2}-\beta^{n+2}}{\alpha-\beta} \biggr) \\ =&\frac{1}{3(\gamma-\delta)} \biggl[ \gamma^{n} \biggl( \frac{6}{\gamma }+\frac{1}{\gamma^{2}}+\gamma^{2} \biggr) +\delta^{n} \biggl( \frac{-6}{\delta }-\frac{1}{\delta^{2}}-\delta^{2} \biggr) \biggr] \\ &{}+\frac{1}{3(\alpha-\beta)} \biggl[ \alpha^{n} \biggl( \frac{-6}{\alpha }-\frac{1}{\alpha^{2}}-\alpha^{2} \biggr) + \beta^{n} \biggl( \frac {6}{\beta}+\frac{1}{\beta^{2}}+ \beta^{2} \biggr) \biggr] \\ =&\bigl(\gamma^{n}+\delta^{n}\bigr)-\bigl( \alpha^{n}+\beta^{n}\bigr), \end{aligned}$$
since \(\frac{6}{\gamma}+\frac{1}{\gamma^{2}}+\gamma ^{2}=\frac{-6}{\delta}-\frac{1}{\delta^{2}}-\delta ^{2}=6\sqrt{2}\) and \(\frac{-6}{\alpha}-\frac{1}{\alpha ^{2}}-\alpha^{2}=\frac{6}{\beta}+ \frac{1}{\beta^{2}}+\beta ^{2}=-3\sqrt{5}\).

(3) Notice that \(F_{n}=\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta}\) and \(P_{n}=\frac{\gamma^{n}-\delta^{n}}{\gamma-\delta}\). So we get \(\sqrt{5}F_{n}=\alpha^{n}-\beta^{n}\) and \(2\sqrt{2} P_{n}=\gamma ^{n}-\delta^{n}\). Thus clearly, \(\sqrt{5}F_{n}+2\sqrt{2}P_{n}=(\gamma^{n}-\delta^{n})+(\alpha ^{n}-\beta^{n})\).

(4) It is easily seen that \(P_{n+1}+P_{n-1}=\gamma^{n}+\delta^{n}\). Also \(L_{n}=\alpha^{n}+\beta^{n}\). So \(L_{n}+P_{n+1}+P_{n-1}=\alpha ^{n}+\beta^{n}+\gamma^{n}+\delta^{n}\).

(5) Since \(W_{n+1}=3W_{n}-3W_{n-2}-W_{n-3}\), we easily get
$$\begin{aligned} W_{n+1}-W_{n} =&2W_{n}-3W_{n-2}-W_{n-3} \\ =&2 \biggl( \frac{\gamma^{n}-\delta^{n}}{\gamma-\delta }-\frac{\alpha^{n}-\beta^{n}}{\alpha-\beta} \biggr) -3 \biggl( \frac{\gamma^{n-2}-\delta ^{n-2}}{\gamma-\delta}-\frac{\alpha^{n-2}-\beta^{n-2}}{\alpha -\beta} \biggr) \\ &{}- \biggl( \frac{\gamma^{n-3}-\delta^{n-3}}{\gamma-\delta }-\frac{\alpha ^{n-3}-\beta^{n-3}}{\alpha-\beta} \biggr) \\ =&\frac{1}{\gamma-\delta} \biggl[ \gamma^{n} \biggl( 2- \frac{3}{\gamma ^{2}}-\frac{1}{\gamma^{3}} \biggr) +\delta^{n} \biggl( -2+\frac{3}{\delta ^{2}}+\frac{1}{\delta^{3}} \biggr) \biggr] \\ &{}+\frac{1}{\alpha-\beta} \biggl[ \alpha^{n-1} \biggl( 2\alpha- \frac{3}{ \alpha}-\frac{1}{\alpha^{2}} \biggr) -\beta^{n-1} \biggl( 2\beta- \frac {3}{\beta}-\frac{1}{\beta^{2}} \biggr) \biggr] \end{aligned}$$
and hence
$$\begin{aligned}& 2W_{n+1}-2W_{n} = \frac{2}{2\sqrt{2}} \biggl[ \gamma^{n} \biggl( \frac{2\gamma^{3}-3\gamma-1}{\gamma^{3}} \biggr) +\delta ^{n} \biggl( \frac{-2\delta ^{2}+3\delta+1}{\delta^{3}} \biggr) \biggr] \\& \hphantom{2W_{n+1}-2W_{n} ={}}{} -\frac{2}{\alpha-\beta} \biggl[ \alpha^{n-1} \biggl( \frac{2\alpha^{3}-3\alpha-1}{\alpha^{2}} \biggr) -\beta ^{n-1} \biggl( \frac{2\beta ^{3}-3\beta-1}{\beta^{2}} \biggr) \biggr] \\& \quad \Leftrightarrow \quad 2W_{n+1}-2W_{n} + \frac{2}{\alpha-\beta} \biggl[ \alpha^{n-1} \biggl( \frac{2\alpha^{3}-3\alpha-1}{\alpha^{2}} \biggr) -\beta ^{n-1} \biggl( \frac{2\beta ^{3}-3\beta-1}{\beta^{2}} \biggr) \biggr] \\& \hphantom{\quad \Leftrightarrow \quad}\quad = \frac{1}{\sqrt{2}} \biggl[ \gamma^{n} \biggl( \frac{2\gamma ^{3}-3\gamma-1}{\gamma^{3}} \biggr) +\delta^{n} \biggl( \frac{-2\delta^{3}+3\delta+1}{\delta^{3}} \biggr) \biggr] \\& \quad \Leftrightarrow\quad 2 ( W_{n+1}-W_{n}+F_{n-1} ) =\gamma ^{n}+\delta^{n}, \end{aligned}$$
since \(\frac{2\gamma^{3}-3\gamma-1}{\gamma^{3}}=\frac{-2\delta ^{3}+3\delta+1}{\delta^{3}}=\sqrt{2}\) and \(\frac{2\alpha ^{3}-3\alpha-1}{\alpha^{2}}=\frac{2\beta^{3}-3\beta-1}{\beta ^{2}}=1\).

(6) It is just an algebraic computation, since \(W_{n}= ( \frac{\gamma^{n}-\delta^{n}}{\gamma-\delta} ) - ( \frac{\alpha^{n}-\beta^{n}}{\alpha-\beta} ) \). □

Theorem 4

The sum of the first n terms of \(W_{n}\) is
$$ \sum_{i=1}^{n}W_{i}= \frac{W_{n}+4W_{n-1}+4W_{n-2}+W_{n-3}+1}{2} $$
(8)
for \(n\geq3\).

Proof

Recall that \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\). So
$$ W_{n-3}+W_{n-4}=3W_{n-1}-2W_{n-3}-W_{n}. $$
(9)
Applying (9), we deduce that
$$ \begin{aligned} &W_{1}+W_{0} = 3W_{3}-2W_{1}-W_{4} , \\ &W_{2}+W_{1} = 3W_{4}-2W_{2}-W_{5} , \\ &W_{3}+W_{2} = 3W_{5}-2W_{3}-W_{6} , \\ & \ldots , \\ &W_{n-4}+W_{n-5} = 3W_{n-2}-2W_{n-4}-W_{n-1} , \\ &W_{n-3}+W_{n-4} = 3W_{n-1}-2W_{n-3}-W_{n}. \end{aligned} $$
(10)
If we sum of both sides of (10), then we obtain \(W_{n-3}+W_{0}+2(W_{1}+\cdots+W_{n-4})=3(W_{3}+W_{4}+\cdots +W_{n-1})-2(W_{1}+W_{2}+\cdots+W_{n-3})-(W_{4}+W_{5}+\cdots +W_{n})\). So we get \(W_{n-3}+2(W_{1}+W_{2}+\cdots +W_{n-4})=1-W_{n-2}-W_{n-1}-W_{n}+3W_{n-2}+3W_{n-1}\) and hence we get the desired result. □

Theorem 5

The recurrence relations are
$$\begin{aligned}& W_{2n} = 9W_{2n-2}-20W_{2n-4}+9W_{2n-6}-W_{2n-8} , \\& W_{2n+1} = 9W_{2n-1}-20W_{2n-3}+9W_{2n-5}-W_{2n-7} \end{aligned}$$
for \(n\geq4\).

Proof

Recall that \(W_{n}=3W_{n-1}-3W_{n-3}-W_{n-4}\). So \(W_{2n}=3W_{2n-1}-3W_{2n-3}-W_{2n-4}\) and hence
$$\begin{aligned} W_{2n} =&3W_{2n-1}-3W_{2n-3}-W_{2n-4} \\ =&9W_{2n-2}-9W_{2n-4}-3W_{2n-5}-9W_{2n-4}+9W_{2n-6}+3W_{2n-7} \\ &{}+W_{2n-8}-W_{2n-8}-W_{2n-4} \\ =&-(3W_{2n-5}-3W_{2n-7}-W_{2n-8})+9W_{2n-2}-18W_{2n-4}+9W_{2n-6} \\ &{}-W_{2n-8}-W_{2n-4} \\ =&-W_{2n-4}+9W_{2n-2}-9W_{2n-4}-9W_{2n-4}+9W_{2n-6}-W_{2n-8}-W_{2n-4} \\ =&9W_{2n-2}-20W_{2n-4}+9W_{2n-6}-W_{2n-8}. \end{aligned}$$
The other assertion can be proved similarly. □
The rank of an integer N is defined to be
$$ \rho(N)=\left \{ \begin{array}{l@{\quad}l} p & \mbox{if }p\mbox{ is the smallest prime with }p|N, \\ \infty& \mbox{if }N\mbox{ is prime}. \end{array} \right . $$
Thus we can give the following theorem.

Theorem 6

The rank of \(W_{n}\) is
$$ \rho(W_{n})=\left \{ \begin{array}{l@{\quad}l} 2 & \textit{if }n=5+6k,6+6k,7+6k, \\ 3 & \textit{if }n=8+12k,9+12k,15+12k,16+12k, \\ 5 & \textit{if }n=14+60k,46+60k \end{array} \right . $$
for an integer \(k\geq0\).

Proof

Let \(n=5+6k\). We prove it by induction on k. Let \(k=0\). Then we get \(W_{5}=24=2^{3}\cdot3\). So \(\rho(W_{5})=2\). Let us assume that the rank of \(W_{n}\) is 2 for \(n=k-1\), that is, \(\rho(W_{6k-1})=2\), so \(W_{5+6(k-1)}=W_{6k-1}=2^{a}\cdot B\) for some integers \(a\geq1\) and \(B>0\). For \(n=k\), we get
$$\begin{aligned} W_{6k+5} =&3W_{6k+4}-3W_{6k+2}-W_{6k+1} \\ =&3(3W_{6k+3}-3W_{6k+1}-W_{6k})-3W_{6k+2}-W_{6k+1} \\ =&9W_{6k+3}-9W_{6k+1}-3W_{6k}-3W_{6k+2}-W_{6k+1} \\ =&9(3W_{6k+2}-3W_{6k}-W_{6k-1})-9W_{6k+1}-3W_{6k}-3W_{6k+2}-W_{6k+1} \\ =&27W_{6k+2}-27W_{6k}-9W_{6k-1}-9W_{6k+1}-3W_{6k}-3W_{6k+2}-W_{6k+1} \\ =&24W_{6k+2}-30W_{6k}-10W_{6k+1}-9W_{6k-1} \\ =&24W_{6k+2}-30W_{6k}-10W_{6k+1}-9 \cdot2^{a}B \\ =&2\bigl[12W_{6k+2}-15W_{6k}-5W_{6k+1}-9 \cdot2^{a-1}B\bigr]. \end{aligned}$$
Therefore \(\rho(W_{5+6k})=2\). Similarly it can be shown that \(\rho(W_{6+6k})=\rho(W_{7+6k})=2\).
Now let \(n=8+12k\). For \(k=0\), we get \(W_{8}=387=3^{2}\cdot43\). So \(\rho(W_{8})=3\). Let us assume that for \(n=k-1\) the rank of \(W_{n}\) is 3, that is, \(\rho(W_{8+12(k-1)})=\rho (W_{12k-4})=3^{b}\cdot H\) for some integers \(b\geq1\) and \(H>0\) which is not even integer. For \(n=k\), we get
$$\begin{aligned} W_{12k+8} =&3W_{12k+7}-3W_{12k+5}-W_{12k+4} \\ =&3W_{12k+7}-3W_{12k+5}-(3W_{12k+3}-3W_{12k+1}-W_{12k}) \\ =&3W_{12k+7}-3W_{12k+5}-3W_{12k+3}+3W_{12k+1}+W_{12k} \\ =&3W_{12k+7}-3W_{12k+5}-3W_{12k+3}+3W_{12k+1} \\ &{}+(3W_{12k-1}-3W_{12k-3}-W_{12k-4}) \\ =&3W_{12k+7}-3W_{12k+5}-3W_{12k+3}+3W_{12k+1}+3W_{12k-1} \\ &{}-3W_{12k-3}-W_{12k-4} \\ =&3W_{12k+7}-3W_{12k+5}-3W_{12k+3}+3W_{12k+1}+3W_{12k-1} \\ &{}-3W_{12k-3}-3^{b}\cdot H \\ =&3\bigl(W_{12k+7}-W_{12k+5}-W_{12k+3}+W_{12k+1}+W_{12k-1} \\ &{}-W_{12k-3}-3^{b-1}\cdot H\bigr). \end{aligned}$$
So \(\rho(W_{12k+8})=3\). The others can be proved similarly. □

Remark 1

Apart from the above theorem, we see that \(\rho(W_{22})=\rho (W_{26})=\infty\), while \(\rho(W_{70})=\rho(W_{98})=13\) and \(\rho (W_{10})=\rho(W_{34})=\rho(W_{50})=23\). But there is no general formula.

The companion matrix for \(W_{n}\) is
$$M=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 3 & 0 & -3 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{array} \right ]. $$
Set
$$N=\left [ \begin{array}{@{}c@{}} 1 \\ 0 \\ 0 \\ 0 \end{array} \right ] $$
and
$$R=[ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 3 & 1 & 0 & 0\end{array} ] . $$
Then we can give the following theorem, which can be proved by induction on n.

Theorem 7

For the sequence \(W_{n}\), we have:

(1) \(RM^{n}N=W_{n+3}+P_{n}+2(W_{n+1}-F_{n})\) for \(n\geq1\).

(2) \(R(M^{T})^{n-3}N=W_{n}\) for \(n\geq3\).

(3) If \(n\geq7\) is odd, then
$$ M^{n}=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24} \\ m_{31} & m_{32} & m_{33} & m_{34} \\ m_{41} & m_{42} & m_{43} & m_{44}\end{array} \right ] , $$
where
$$\begin{aligned}& m_{11} =W_{n+2},\qquad m_{21}=W_{n+1}, \qquad m_{31}=W_{n},\qquad m_{41}=W_{n-1} , \\& m_{14} =-W_{n+1},\qquad m_{24}=-W_{n}, \qquad m_{34}=-W_{n-1},\qquad m_{44}=-W_{n-2}, \\& m_{12} =-1-W_{n+1}-2\sum_{i=0}^{\frac {n-5}{2}}W_{n-1-2i}, \qquad m_{13}=-W_{n+2}-2\sum _{i=0}^{\frac{n-3}{2}}W_{n-2i} , \\& m_{22} =-W_{n}-2\sum_{i=0}^{\frac{n-5}{2}}W_{n-2-2i}, \qquad m_{23}=-1-W_{n+1}-2\sum _{i=0}^{\frac{n-5}{2}}W_{n-1-2i}, \\& m_{32} =-1-W_{n-1}-2\sum_{i=0}^{\frac{n-7}{2}}W_{n-3-2i}, \qquad m_{33}=-W_{n}-2\sum _{i=0}^{\frac{n-5}{2}}W_{n-2-2i}, \\& m_{42} =-W_{n-2}-2\sum_{i=0}^{\frac{n-7}{2}}W_{n-4-2i}, \qquad m_{43}=-1-W_{n-1}-2\sum _{i=0}^{\frac{n-7}{2}}W_{n-3-2i}, \end{aligned}$$
and if \(n\geq8\) is even, then
$$ M^{n}=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} m_{11} & m_{12} & m_{13} & m_{14} \\ m_{21} & m_{22} & m_{23} & m_{24} \\ m_{31} & m_{32} & m_{33} & m_{34} \\ m_{41} & m_{42} & m_{43} & m_{44}\end{array} \right ] , $$
where
$$\begin{aligned}& m_{11} =W_{n+2},\qquad m_{21}=W_{n+1}, \qquad m_{31}=W_{n},\qquad m_{41}=W_{n-1}, \\& m_{14} =-W_{n+1},\qquad m_{24}=-W_{n}, \qquad m_{34}=-W_{n-1},\qquad m_{44}=-W_{n-2}, \\& m_{12} =-W_{n+1}-2\sum_{i=0}^{\frac {n-4}{2}}W_{n-1-2i}, \qquad m_{13}=-1-W_{n+2}-2\sum _{i=0}^{\frac{n-4}{2}}W_{n-2i}, \\& m_{22} =-1-W_{n}-2\sum_{i=0}^{\frac{n-6}{2}}W_{n-2-2i}, \qquad m_{23}=-W_{n+1}-2\sum _{i=0}^{\frac{n-4}{2}}W_{n-1-2i}, \\& m_{32} =-W_{n-1}-2\sum_{i=0}^{\frac{n-6}{2}}W_{n-3-2i}, \qquad m_{33}=-1-W_{n}-2\sum _{i=0}^{\frac{n-6}{2}}W_{n-2-2i}, \\& m_{42} =-1-W_{n-2}-2\sum_{i=0}^{\frac{n-8}{2}}W_{n-4-2i}, \qquad m_{43}=-W_{n-1}-2\sum _{i=0}^{\frac{n-6}{2}}W_{n-3-2i}. \end{aligned}$$
A circulant matrix is a matrix \(A=[a_{ij}]_{n\times n}\) defined to be
$$ A=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{0} & a_{1} & a_{2} & \cdots& a_{n-1} \\ a_{n-1} & a_{0} & a_{1} & \cdots& a_{n-2} \\ a_{n-2} & a_{n-1} & a_{0} & \cdots& a_{n-3} \\ \cdot&\cdot&\cdot&\cdots&\cdot \\ \cdot&\cdot&\cdot&\cdots&\cdot \\ a_{1} & a_{2} & a_{3} & \cdots& a_{0}\end{array} \right ] , $$
where \(a_{i}\) are constants. The eigenvalues of A are
$$ \lambda_{j}(A)=\sum_{k=0}^{n-1}a_{k}w^{-jk}, $$
(11)
where \(w=e^{\frac{2\pi i}{n}}\), \(i=\sqrt{-1}\), and \(j=0,1,\ldots ,n-1\). The spectral norm for a matrix \(B=[b_{ij}]_{n\times m}\) is defined to be \(\|B\|_{\mathrm{spec}}=\max\{\sqrt{\lambda_{i}}\}\), where \(\lambda_{i}\) are the eigenvalues of \(B^{H}B\) for \(0\leq j\leq n-1\) and \(B^{H}\) denotes the conjugate transpose of B.
For the circulant matrix
$$ W=W(W_{n})=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} W_{0} & W_{1} & W_{2} & \cdots& W_{n-1} \\ W_{n-1} & W_{0} & W_{1} & \cdots& W_{n-2} \\ W_{n-2} & W_{n-1} & W_{0} & \cdots& W_{n-3} \\ \cdot&\cdot&\cdot&\cdots&\cdot \\ \cdot&\cdot&\cdot&\cdots&\cdot \\ W_{1} & W_{2} & W_{3} & \cdots& W_{0}\end{array} \right ] $$
for \(W_{n}\), we can give the following theorem.

Theorem 8

The eigenvalues of W are
$$ \lambda_{j}(W)=\frac{\left \{ \begin{array}{@{}l@{}} W_{n-1}w^{-3j}+(W_{n}+P_{n-1}-2F_{n-1}+1)w^{-2j} \\ \quad {}+(P_{n}-2F_{n}-W_{n-1})w^{-j}-W_{n}\end{array} \right \} } {w^{-4j}+3w^{-3j}-3w^{-j}+1} $$
for \(j=0,1,2,\ldots,n-1\).

Proof

Applying (11) we easily get
$$\begin{aligned} \lambda_{j}(W) =&\sum_{k=0}^{n-1}W_{k}w^{-jk} =\sum_{k=0}^{n-1} \biggl( \frac{\gamma^{k}-\delta^{k}}{\gamma-\delta }-\frac{\alpha^{k}-\beta^{k}}{\alpha-\beta} \biggr) w^{-jk} \\ =&\frac{1}{\gamma-\delta} \biggl[ \frac{\gamma^{n}-1}{\gamma w^{-j}-1}-\frac{\delta^{n}-1}{\delta w^{-j}-1} \biggr] -\frac{1}{\alpha -\beta} \biggl[ \frac{\alpha^{n}-1}{\alpha w^{-j}-1}-\frac{\beta^{n}-1}{\beta w^{-j}-1} \biggr] \\ =&\frac{1}{\gamma-\delta} \biggl[ \frac{(\gamma^{n}-1)(\delta w^{-j}-1)-(\delta^{n}-1)(\gamma w^{-j}-1)}{(\gamma w^{-j}-1)(\delta w^{-j}-1)} \biggr] \\ &{}-\frac{1}{\alpha-\beta} \biggl[ \frac{(\alpha^{n}-1)(\beta w^{-j}-1)-(\beta^{n}-1)(\alpha w^{-j}-1)}{(\alpha w^{-j}-1)(\beta w^{-j}-1)} \biggr] \\ =&\frac{1}{\gamma-\delta} \biggl[ \frac{w^{-j}(\gamma^{n}\delta -\delta^{n}\gamma+\gamma-\delta)+\delta^{n}-\gamma ^{n}}{\delta\gamma w^{-2j}-w^{-j}(\delta+\gamma)+1} \biggr] \\ &{}-\frac{1}{\alpha-\beta} \biggl[ \frac{w^{-j}(\alpha^{n}\beta -\beta^{n}\alpha+\alpha-\beta)+\beta^{n}-\alpha^{n}}{\beta \alpha w^{-2j}-w^{-j}(\beta+\alpha)+1} \biggr] \\ =&\frac{\left \{ \begin{array}{@{}l@{}} w^{-3j}[\sqrt{5}(\delta-\gamma+\gamma\delta^{n}-\delta\gamma ^{n})+2\sqrt{2}(\alpha-\beta+\alpha^{n}\beta-\alpha\beta^{n})] \\ \quad {}+w^{-2j}[\sqrt{5}(\gamma^{n}-\delta^{n}+\delta-\gamma+\gamma \delta ^{n}-\gamma^{n}\delta)+2\sqrt{2}(\beta^{n}-\alpha^{n}) \\ \quad {}+4\sqrt{2}(\alpha-\beta+\alpha^{n}\beta-\alpha\beta ^{n})]+w^{-j}[\sqrt{5}(\gamma^{n}-\delta^{n}+\gamma-\delta \\ \quad {}+\gamma^{n}\delta-\gamma\delta^{n})+2\sqrt{2}(\beta-\alpha +\beta^{n}\alpha-\alpha^{n}\beta)+4\sqrt{2}(\beta^{n}-\alpha^{n})] \\ \quad {}+[\sqrt{5}(\delta^{n}-\gamma^{n})+2\sqrt{2}(\alpha^{n}-\beta^{n})]\end{array} \right \} } {2\sqrt{10}(w^{-4j}+3w^{-3j}-3w^{-j}+1)} \\ =&\frac{\left \{ \begin{array}{@{}l@{}} W_{n-1}w^{-3j}+(W_{n}+P_{n-1}-2F_{n-1}+1)w^{-2j} \\ \quad {}+(P_{n}-2F_{n}-W_{n-1})w^{-j}-W_{n}\end{array} \right \} } {w^{-4j}+3w^{-3j}-3w^{-j}+1}, \end{aligned}$$
since \(\alpha\beta=-1\), \(\gamma\delta=-1\), \(\alpha+\beta=1\), \(\alpha-\beta=\sqrt{5}\), \(\gamma+\delta=2\), and \(\gamma-\delta =2\sqrt{2}\). □
After all, we consider the spectral norm of W. Let \(n=2\). Then \(W_{2}=[0]_{2\times2}\). So \(\|W_{2}\|_{\mathrm{spec}}=0\). Similarly for \(n=3\), we get
$$ W_{3}=\left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array} \right ] $$
and hence \(W_{3}^{H}W_{3}=I_{3}\). So \(\|W_{3}\|_{\mathrm{spec}}=1\). For \(n\geq4\), the spectral norm of \(W_{n}\) is given by the following theorem, which can be proved by induction on n.

Theorem 9

The spectral norm of \(W_{n}\) is
$$ \|W_{n}\|_{\mathrm{spec}}=\frac{W_{n-1}+4W_{n-2}+4W_{n-3}+W_{n-4}+1}{2} $$
for \(n\geq4\).
For example, let \(n=6\). Then the eigenvalues of \(W_{6}^{H}W_{6}\) are
$$ \lambda_{0}=1\text{,}369,\qquad \lambda_{1}=289, \qquad \lambda_{2}=\lambda _{4}=784 \quad \mbox{and} \quad \lambda_{3}=\lambda_{5}=388. $$
So the spectral norm is \(\|W_{6}\|_{\mathrm{spec}}=\sqrt{\lambda_{0}}=37\). Also \(\frac{W_{5}+4W_{4}+4W_{3}+W_{2}+1}{2}=37\). Consequently,
$$ \|W_{6}\|_{\mathrm{spec}}=\frac{W_{5}+4W_{4}+4W_{3}+W_{2}+1}{2}=37 $$
as we claimed.

Declarations

Acknowledgements

The author wishes to thank Professor Ahmet Tekcan of Uludag University for constructive suggestions.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Arts and Science, Düzce University

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© Özkoç; licensee Springer. 2015