\documentclass[12pt,reqno]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{epsf}
\usepackage{float}
\usepackage[usenames]{color}
\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}
\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}
\newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
%\newenvironment{proof}{\textit{Proof:} \ignorespaces}{\nopagebreak\null\hfill\hbox{$\Box$}}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}
\begin{document}
\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\mbox{\null}
\hspace*{4cm}
\end{center}
\begin{center}
\vskip 1cm{\LARGE\bf Quasi-Fibonacci Numbers of the \\
\vskip .1in
Seventh Order}
\vskip 1cm
\large
Roman Witu{\l}a, Damian S{\l}ota and Adam Warzy{\'n}ski \\
Institute of Mathematics \\
Silesian University of Technology \\
Kaszubska 23 \\
Gliwice 44-100 \\
Poland \\
\href{mailto:r.witula@polsl.pl}{\tt r.witula@polsl.pl} \\
\href{mailto:d.slota@polsl.pl}{\tt d.slota@polsl.pl} \\
\end{center}
\vskip .2in
\begin{abstract}
In this paper we introduce and investigate
the so-called quasi-Fibonacci numbers of the seventh
order. We discover
many surprising relations
and identities, and study some applications to polynomials.
\end{abstract}
\vskip .2in
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% Section 1 %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Grzymkowski and Witu{\l}a~\cite{grzymkowski}
discovered and studied
the following two identities:
\begin{align}
(1+\xi +\xi ^4)^n&=F_{n+1}+F_n(\xi +\xi ^4),\label{nw-g1}\\
(1+\xi ^2+\xi ^3)^n&=F_{n+1}+F_n(\xi ^2+\xi ^3),\label{nw-g2}
\end{align}
where $F_n$ denote the Fibonacci numbers and $\xi\in\mathbb{C}$ is
a~primitive fifth root of unity (i.e.,~$\xi^5=1$ and
$\xi\neq 1$). These identities make it possible to prove many
classical relations for Fibonacci numbers as well as to generalize
some of them. We may state that these identities make up an
independent method of proving such relations, which is an
alternative to the methods depending on the application of either
Binet formulas or the generating function of Fibonacci and Lucas
numbers.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example of the application of identities~(\ref{nw-g1}) and~(\ref{nw-g2})}
First, we note that
\begin{equation}\label{wzor-a}
(1+\xi +\xi^4)^{u+v}=F_{u+v+1}+ F_{u+v} \, \big( \xi + \xi^4\big)
\end{equation}
and
\begin{multline}\label{wzor-b}
(1+\xi +\xi^4)^{u+v}=(1+\xi +\xi^4)^u\,(1+\xi +\xi^4)^v= \\
=\big( F_{u+1}+F_u\,(\xi +\xi^4)\big)\,\big( F_{v+1}+F_v\,(\xi +\xi^4)\big)= \\
\mbox{(by the identity $1+\xi+\xi^2+\xi^3+\xi^4=0$)}\\
=F_{u+1}\,F_{v+1}+F_u\,F_v+(F_u\,F_{v+1}+F_{u+1}\,F_v-F_u\,F_v)\,(\xi +\xi ^4)=\\
=F_{u+1}\,F_{v+1}+F_u\,F_v+(F_u\,F_{v+1}+F_{u-1}\,F_v)\,(\xi +\xi ^4).
\end{multline}
Replacing $u$ by $u-r$ and $v$ by $v+r$ in~(\ref{wzor-b}) we obtain
\begin{multline}\label{wzor-c}
(1+\xi +\xi^4)^{u+v}=F_{u-r+1}\,F_{v+r+1}+F_{u-r}\,F_{v+r}+ \\
+(F_{u-r}\,F_{v+r+1}+F_{u-r+1}\,F_{v+r}-F_{u-r}\,F_{v+r})(\xi +\xi ^4).
\end{multline}
We note that the numbers~$1$ and $\xi+\xi^4$ are linearly independent over~$\mathbb{Q}$.
Hence comparing the parts without $(\xi +\xi^4)$ of (\ref{wzor-a}) with (\ref{wzor-b})
and (\ref{wzor-b}) with (\ref{wzor-c}) we get two known identities
(see~\cite{graham,koshy})
\[
F_{u+v+1} = F_{u+1}\, F_{v+1} + F_{u}\, F_{v}
\]
and
\begin{align*}
F_{u+1}\,F_{v+1}-F_{u-r+1}\,F_{v+r+1}&=F_{u-r}\,F_{v+r}-F_u\,F_v \\
&\mbox{(after the next $(u-r)$ iterations)} \\
&=(-1)^{u-r+1}(F_r\,F_{v-u+r}-F_0\,F_{v-u+2r}) \\
&=(-1)^{u-r+1}F_r\,F_{v-u+r}.
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The aim of the paper}
In this paper the relations~(\ref{nw-g1}) and~(\ref{nw-g2}) will
be generalized in the following way:
\begin{align*}
(1+\delta (\xi+\xi^6))^n &=
A_n(\delta)+B_n(\delta)(\xi+\xi^6)+C_n(\delta)(\xi^2+\xi^5),\\
(1+\delta (\xi^2 +\xi^5))^n&= A_n(\delta )+B_n(\delta )
(\xi^2+\xi^5)+C_n(\delta )(\xi^3 +\xi^4),
\end{align*}
and
$$
(1+\delta (\xi^3 +\xi^4))^n =
A_n(\delta )+B_n(\delta )(\xi^3 +\xi^4) +C_n(\delta )(\xi +\xi^6),
$$
where $\xi\in\mathbb{C}$
are primitive seventh roots of unity (i.e.,~$\xi^7=1$ and $\xi\neq 1$),
$\delta\in \mathbb{C}$, $\delta\neq 0$.
New families of numbers created by these identities
\begin{equation}\label{pierwszy}
\{A_{n}(\delta) \}_{n=1}^{\infty},\qquad
\{B_{n}(\delta) \}_{n=1}^{\infty},\quad \mbox{and}\quad
\{C_{n}(\delta) \}_{n=1}^{\infty}
\end{equation}
called here ``the quasi-Fibonacci numbers of order $(7;\delta)$'',
$\delta\in \mathbb{C}$, $\delta \neq 0$, are investigated in this
paper. The elements of each of the three
sequences~(\ref{pierwszy}) satisfy the same recurrence relation of
order three
$$
\mathbb{X}_{n+3}+(\delta -3)\mathbb{X}_{n+2}+(3-2\delta
-2\delta^2)\mathbb{X}_{n+1} + (-1+\delta
+2\delta^2-\delta^3)\mathbb{X}_n=0,
$$
which enables a~direct trigonometrical
representation of these numbers (see formulas~(\ref{2.14})--(\ref{2.16})).
In consequence, many surprising
algebraic and trigonometric identities
and summation formulas for these numbers may be generated.
Also the polynomials connected with the numbers $\{C_{n}(\delta) \}_{n=1}^{\infty}$
$$
\sum_{k=1}^{n} C_{k+1}(\delta)\, x^{n-k},\qquad n\in \mathbb{N},
$$
are investigated in this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% Section 2 %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Minimal polynomials, linear independence over~$\mathbb{Q}$}
Let $\Psi _n(x)$ be the minimal polynomial of $\cos (2\pi /n)$ for every $n\in\mathbb{N}$.
W.~Watkins and J.~Zeitlin described \cite{watkins}
(see also~\cite{surowski}) the following identities:
$$
T_{s+1}(x)-T_s(x)=2^s\prod\limits_{d|n}\Psi _d(x)
$$
if $n=2s+1$ and
$$
T_{s+1}(x)-T_{s-1}(x)=2^s\prod\limits_{d|n}\Psi _d(x)
$$
if $n=2s$, where $T_s(x)$ denotes the $s$-th Chebyshev polynomial of the first kind.
In the sequel, if $n=2s+1$ is a prime number, we obtain
$$
T_{s+1}(x)-T_s(x)=2^s\Psi _1(x)\Psi _n(x).
$$
For example, we have
\begin{align*}
\Psi_7(x) &= \frac{1}{8(x-1)}(T_4(x)-T_3(x)) =\frac{1}{8(x-1)}(8x^4-4x^3-8x^2+3x+1) \\
&=\frac{1}{8(x-1)}8(x-1)\bigg( x^3+\frac{1}{2}x^2-\frac{1}{2}x-\frac{1}{8}\bigg) =
x^3+\frac{1}{2}x^2-\frac{1}{2}x-\frac{1}{8},
\end{align*}
\begin{lemma}\label{qf2-lem1.1}
If $n\geqslant 3$ then the roots of $\Psi _n(x)=0$ are $\cos (2\pi k/n)$,
for $0