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\begin{center}
\vskip 1cm{\LARGE\bf On the Log-Concavity of the Hyperfibonacci\\
\vskip .10in
Numbers and the Hyperlucas Numbers} \vskip 1cm
\large Li-Na Zheng and Rui Liu \\
School of Mathematical Sciences\\
Dalian University of Technology \\
Dalian 116024  \\
P. R. China\\
\href{mailto:linazheng2007@yahoo.com.cn}{\tt linazheng2007@yahoo.com.cn}\\
\ \\
Feng-Zhen Zhao\\
Department of Mathematics\\
Shanghai University\\
Shanghai 200444\\
P. R. China\\
\href{mailto:fengzhenzhao@shu.edu.cn}{\tt fengzhenzhao@shu.edu.cn}\\
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\begin{abstract}
In this paper, we discuss the properties of the hyperfibonacci
numbers $F^{[r]}_n$ and hyperlucas numbers $L^{[r]}_n$. We
investigate the log-concavity (log-convexity) of hyperfibonacci
numbers and hyperlucas numbers. For example, we prove that
$\{F^{[r]}_n\}_{n\ge 1}$ is log-concave. In addition, we also study
the log-concavity (log-convexity) of generalized hyperfibonacci
numbers and hyperlucas numbers.
\end{abstract}

\section{Introduction}
Let $\{F_n\}_{n\ge 0}$ and $\{L_n\}_{n\ge 0}$ denote the Fibonacci
and Lucas sequence, respectively. It is well known that the Binet
forms of $F_n$ and $L_n$ are
\begin{eqnarray}
F_n=\frac{\alpha^n-(-1)^n\alpha^{-n}}{\sqrt 5}, \quad
L_n=\alpha^n+(-1)^n\alpha^{-n}, \label{ad-1.1}
\end{eqnarray}
where $\alpha=(1+\sqrt 5)/2$. It is evident that $\{F_n\}_{n\ge 0}$
and $\{L_n\}_{n\ge 0}$ satisfy
\begin{eqnarray}
W_n=W_{n-1}+W_{n-2}, \quad n\ge 2. \label{ad-1.2}
\end{eqnarray}
For positive integers $r$, hyperfibonacci numbers $F_n^{[r]}$ and
hyperlucas numbers $L_n^{[r]}$ are defined as follows \cite{ref5}:
\begin{eqnarray*}
F_n^{[r]}=\sum_{k=0}^nF_k^{[r-1]}, \quad
L_n^{[r]}=\sum_{k=0}^nL_k^{[r-1]},
\end{eqnarray*}
where $F_n^{[0]}=F_n$ and $L_n^{[0]}=L_n$. Initial values of $\{F_n^{[1]}\}$ and $\{L_n^{[1]}\}$ are as follows:\\
\begin{center}
\begin{tabular}{c|ccccccccccccccc}
\hline {\em n} &amp; $0$ &amp; $1$ &amp; $2$ &amp;
 $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$ &amp; $10$ &amp; $11$ &amp; $12$ &amp; $13$ &amp; $14$ \\
  \hline
\\
  $F_n^{[1]}$ &amp; $0$ &amp; $1$ &amp; $2$ &amp;
  $4$ &amp; $7$ &amp; $12$ &amp; $20$ &amp; $33$ &amp; $54$ &amp; $88$ &amp; $143$ &amp; $232$ &amp; $376$ &amp; $609$ &amp; $986$ \\
  \hline
\\
 $L_n^{[1]}$ &amp; $2$ &amp; $3$ &amp; $6$ &amp; $10$ &amp; $17$ &amp; $28$ &amp; $46$ &amp; $75$ &amp; $122$ &amp; $198$ &amp; $321$ &amp; $520$ &amp; $842$ &amp; $1363$ &amp; $2206$ \\
  \hline
\end{tabular}
\end{center}
In \cite{ref3, ref5, ref9}, some properties of hyperfibonacci
numbers $F_n^{[r]}$ and hyperlucas numbers $L_n^{[r]}$ are given. In
this paper, we continue discussing the properties of $F_n^{[r]}$ and
$L_n^{[r]}$. Now we recall some other definitions involved in this
paper.
\begin{definition}
Let $\{a_n\}_{n\geq 0}$ be a
sequence of positive numbers. If for all $j\geq 1$, $a_j^2\geq
a_{j-1}a_{j+1}$ (respectively $a_{j-1}a_{j+1}\geq a_j^2$), the
sequence $\{a_n\}_{n\geq 0}$ is called log-concave (respectively
log-convex).
\end{definition}
\begin{definition} {\rm\cite{ref6}}
Let $\{a_n\}_{n\geq 0}$ be a sequence of positive real numbers.
We say that $\{a_n\}_{n\geq 0}$ is {\it log-balanced} if $\{a_n\}_{n\ge 0}$
is log-convex and $\{\frac{a_n}{n!}\}_{n\ge 0}$ is log-concave.
\end{definition}
The log-convexity and log-concavity are important properties of
combinatorial sequences, and they play an important role in many
fields such as quantum physics, white noise theory, probability,
economics and mathematical biology. See for instance \cite{ref1,
ref2, ref6, ref7, ref8, ref10, ref11}. Clearly, log-balancedness
implies log-convexity. It is well known that $\{a_n\}_{n\ge 0}$ is
log-convex (log-concave) if and only if its quotient sequence
$\{a_{n+1}/a_n\}_{n\geq 0}$ is nondecreasing (nonincreasing).
Naturally, the quotient sequence of a log-balanced sequence does not
grow too fast. For the Fibonacci sequence $\{F_n\}$ and Lucas
sequence $\{L_n\}$, their log-concavity (log-convexity) are related
to the parity of $n$. It is well known that $\{F_{2n+1}\}$ and
$\{L_{2n}\}$ are log-convex and $\{F_{2n}\}$ and $\{L_{2n+1}\}$ are
log-concave. In this paper, we discuss the log-concavity 
(log-convexity) of hyperfibonacci numbers $F_n^{[r]}$ and hyperlucas
numbers $L_n^{[r]}$. In Section 2, we show that
$\{F_n^{[r]}\}_{n\geq 1}$ and $\{L_n^{[r]}\}_{n\geq 3}$ are
log-concave for $r\ge 1$. In addition, we also consider  the
log-concavity ( log-convexity) of generalized hyperfibonacci numbers
and generalized hyperlucas numbers.

\section{The log-concavity of hyperfibonacci numbers and hyperlucas numbers}
In this section, we state and prove the main results of this paper.
\begin{lemma}\label{lemma:1a}
{\rm\cite{ref4, ref12}} If the sequences $\{x_n\}$ and $\{y_n\}$ are
log-concave, then so is their ordinary convolution
$z_n=\sum_{k=0}^nx_ky_{n-k}, n=0, 1, 2, \ldots.$
\end{lemma}
We note that above $\{F_n^{[r]}\}_{n\geq 1}$ is the convolution of
$\{F_n^{[r]}\}_{n\geq 1}$ and $\{1\}_{n\geq 0}$.
%Well-known results follow immediately from (1.1).
\begin{lemma}\label{lemma:1b}
For $n\ge 0$, the following equalities hold:
\begin{eqnarray*}
L_{n+2}^2-L_{n+1}L_{n+3}=5(-1)^n,    \quad
F_{n+2}^2-F_{n+1}F_{n+3}=(-1)^{n+1}
\end{eqnarray*}
\end{lemma}
\begin{proof}
From (\ref{ad-1.1}), we can immediately prove that this lemma holds.
\end{proof}
\begin{theorem}\label{thm:1a}
For $r\geq 1$, the sequences $\{F_n^{[r]}\}_{n\geq 1}$,
$\{L_n^{[1]}\}_{n\geq 3}$, and $\{L_n^{[r]}\}_{n\geq 0} (r\geq 2)$
are log-concave.
\end{theorem}
\begin{proof} A simple induction using the defining recurrence (\ref{ad-1.2}) shows that
\begin{eqnarray}
F_n^{[1]}=F_{n+2}-1, \quad  L_n^{[1]}=L_{n+2}-1. \label{ad-2.1}
\end{eqnarray}
By using (\ref{ad-2.1}) and (\ref{ad-1.2}), we can verify that
$\{F_n^{[1]}\}$ and $\{L_n^{[1]}\}$ satisfy the following recurrence
relation
\begin{eqnarray}
W_{n+1}=2W_n-W_{n-2}, \quad n\ge 2. \label{ad-2.2}
\end{eqnarray}
For $n\ge 1$, let $x_n=F_{n+1}^{[1]}/F_n^{[1]}$. We prove by
induction that $\{x_n\}_{n\ge 1}$ is decreasing. Clearly,
$x_1=x_2=2&gt;x_3=7/4$. For $n\ge 3$, assume that $x_{k-1}\ge x_k$ for
all 1$\leq$ k $\leq$ n. It follows from (\ref{ad-2.2}) that
\begin{eqnarray*}
x_n=2-\frac{1}{x_{n-1}x_{n-2}}.
\end{eqnarray*}
Then we have
\begin{eqnarray*}
x_n-x_{n+1}=\frac{x_{n-2}-x_n}{x_{n-2}x_{n-1}x_n}.
\end{eqnarray*}
Since $x_{n-2}\ge x_{n-1}\geq x_{n}$, it follows that $x_{n}\geq
x_{n+1}$. Then $\{x_n\}_{n\ge 1}$ is decreasing and
$\{F_n^{[1]}\}_{n\geq 1}$ is log-concave. For $\{L_n^{[1]}\}_{n\geq
0}$, using a similar method, we can prove that $\{L_n^{[1]}\}_{n\geq
3}$ is log-concave. It is clear that the initial cases ($0\le n\le
2$) of $\{L_n^{[1]}\}_{n\geq 0}$ are not log-concave. According to
Lemma \ref{lemma:1a}, we know that $\{F_n^{[r]}\}_{n\geq 1}$ is
log-concave. Now we show that $\{L_n^{[2]}\}_{n\geq 0}$ is
log-concave. We can verify that\begin{eqnarray}
L^{[2]}_n=L_{n+4}-n-5. \label{ad-2.3}
\end{eqnarray}
By using (\ref{ad-2.3}), (\ref{ad-1.2}) and Lemma \ref{lemma:1b}, we
get
\begin{eqnarray*}
\bigg(L^{[2]}_n\bigg)^2-L^{[2]}_{n-1}L^{[2]}_{n+1}&amp;=&amp;(L_{n+4}-n-5)^2-(L_{n+3}-n-4)(L_{n+5}-n-6)\\
&amp;=&amp;L^2_{n+4}-L_{n+3}L_{n+5}+nL_{n+1}+2L_{n-1}+1\\
&amp;=&amp;5(-1)^n+nL_{n+1}+2L_{n-1}+1.
\end{eqnarray*}
When $n\geq 1$,
\begin{eqnarray*}
\bigg(L^{[2]}_n\bigg)^2-L^{[2]}_{n-1}L^{[2]}_{n+1}&amp;\geq&amp;3n-2&gt;0.
\end{eqnarray*}
Then $\{L_n^{[2]}\}_{n\geq 0}$ is log-concave. According to Lemma
\ref{lemma:1a}, we know that $\{L_n^{[r]}\}_{n\geq 0} (r\geq 3)$ is
log-concave.
\end{proof}
\begin{theorem} The sequences $\{n!F_n^{[1]}\}_{n\ge1}$ and $\{n!L_n^{[1]}\}_{n\ge 3}$ are log-balanced.
\end{theorem}
\begin{proof} By Theorem \ref{thm:1a}, in order to prove the log-balancedness of $\{n!F_n^{[1]}\}_{n\ge1}$ and $\{n!L_n^{[1]}\}_{n\ge 3}$, we only need to show that they are log-convex. It follows from (\ref{ad-2.1}), Lemma \ref{lemma:1b} and (\ref{ad-1.2}) that
\begin{eqnarray}
\bigg(F_n^{[1]}\bigg)^2-F_{n-1}^{[1]}F_{n+1}^{[1]}&amp;=&amp;(-1)^{n+1}+F_{n-1}, \label{ad-2.5} \\
\bigg(L_n^{[1]}\bigg)^2-L_{n-1}^{[1]}L_{n+1}^{[1]}&amp;=&amp;5(-1)^n+L_{n-1}.
\label{ad-2.6}
\end{eqnarray}
From (\ref{ad-2.1}), (\ref{ad-2.5}) and (\ref{ad-2.6}), we have
\begin{eqnarray*}
n\bigg(F_n^{[1]}\bigg)^2-(n+1)F_{n-1}^{[1]}F_{n+1}^{[1]}&amp;=&amp;(n+1)[(-1)^{n+1}+F_{n-1}]-(F_{n+2}-1)^2,\\
n\bigg(L_n^{[1]}\bigg)^2-(n+1)L_{n-1}^{[1]}L_{n+1}^{[1]}&amp;=&amp;(n+1)[5(-1)^n+L_{n-1}]-(L_{n+2}-1)^2.
\end{eqnarray*}
Let $S_n=(n+1)[(-1)^{n+1}+F_{n-1}]-(F_{n+2}-1)^2,\quad
T_n=(n+1)[5(-1)^n+L_{n-1}]-(L_{n+2}-1)^2$. Clearly, $S_k&lt;0$ for
$2\le k\le 5$ and $T_k&lt;0$ for $k=4$ or $k=5$. We can prove by
induction that
\begin{eqnarray}
F_n&amp;\ge&amp; n, \quad n\ge5, \label{ad-2.7} \\
L_n&amp;\ge&amp;n, \quad n\ge 0. \label{ad-2.8}
\end{eqnarray}
For $n\ge 6$, by applying (\ref{ad-2.7}), (\ref{ad-2.8}) and
(\ref{ad-1.2}), we obtain
\begin{eqnarray*}
S_n&amp;\le&amp;(n+1)(1+F_{n-1})-(n+1)(F_{n+2}-1)\\
&amp;=&amp;(n+1)(2-2F_n)\\
&amp;&lt;&amp;0,\\
T_n&amp;\le&amp;(n+1)(6-2L_n)\\
&amp;&lt;&amp;0.
\end{eqnarray*}
Noting that
\begin{eqnarray*}
\bigg(n!F_n^{[1]}\bigg)^2-(n-1)!(n+1)!F_{n-1}^{[1]}F_{n+1}^{[1]}&amp;=&amp;(n-1)!n!S_n, \\
\bigg(n!L_n^{[1]}\bigg)^2-(n-1)!(n+1)!L_{n-1}^{[1]}L_{n+1}^{[1]}&amp;=&amp;(n-1)!n!T_n,
\end{eqnarray*}
we have
\begin{eqnarray*}
\bigg(n!F_n^{[1]}\bigg)^2-F_{n-1}^{[1]}F_{n+1}^{[1]}&lt;0\quad for \ \
n\ge 2; \quad
\bigg(n!L_n^{[1]}\bigg)^2-L_{n-1}^{[1]}L_{n+1}^{[1]}&lt;0\quad for \ \
n\ge 4.
\end{eqnarray*}
Hence $\{n!F_n^{[1]}\}_{n\ge1}$ and $\{n!L_n^{[1]}\}_{n\ge 3}$ are
log-convex. As the sequences $\{F_n^{[1]}\}_{n\geq 1}$ and
$\{L_n^{[1]}\}_{n\geq 3}$ are log-concave, the sequences
$\{n!F_n^{[1]}\}_{n\ge1}$ and $\{n!L_n^{[1]}\}_{n\ge 3}$ are
log-balanced.
\end{proof}

In the final part of this section, we discuss the log-concavity
(log-convexity) of generalized hyperfibonacci numbers and hyperlucas
numbers. Let $\{U_n\}_{n\ge 0}$ and $\{V_n\}_{n\ge 0}$ denote the
generalized Fibonacci and Lucas sequence, respectively. Their Binet
forms of $U_n$ and $V_n$ are
\begin{eqnarray}
U_n=\frac{\tau^n-(-1)^n\tau^{-n}}{\sqrt{\Delta}}, \quad
V_n=\tau^n+(-1)^n\tau^{-n}, \label{ad-2.9}
\end{eqnarray}
where $\Delta=p^2+4, \tau=(p+\sqrt{\Delta})/2$, and $p\ge1$. It is
clear that (\ref{ad-2.9}) is a generalization of (\ref{ad-1.1}), and
$\{U_n\}_{n\ge 0}$ and $\{V_n\}_{n\ge 0}$ satisfy the recurrence
\begin{eqnarray}
W_{n+1}=pW_n+W_{n-1}\quad(n\ge1),\quad with\quad U_0=0, U_1=1,
V_0=2, V_1=p.   \label{ad-2.10}
\end{eqnarray}
For positive integers $r$, the generalized hyperfibonacci numbers
$U_n^{[r]}$ and generalized hyperlucas numbers $V_n^{[r]}$ are
defined as follows:
\begin{eqnarray*}
U_n^{[r]}=\sum_{k=0}^nU_k^{[r-1]}, \quad
V_n^{[r]}=\sum_{k=0}^nV_k^{[r-1]},
\end{eqnarray*}
where $U_n^{[0]}=U_n$ and $V_n^{[0]}=V_n$. For some properties of
$\{U_n^{[r]}\}$ and $\{V_n^{[r]}\}$, see \cite{ref3, ref9}. Now we
prove the log-concavity of $\{U_n^{[r]}\}$ and $\{V_n^{[r]}\}$.
\begin{theorem} For $r\geq 1$ and $p\geq 1$, the sequences $\{U_n^{[r]}\}_{n\geq 1}$ and $\{V_n^{[1]}\}_{n\geq 3}$ are log-concave.
\end{theorem}
\begin{proof} From the definitions of $\{U_n^{[1]}\}$ and $\{V_n^{[1]}\}$ and (\ref{ad-2.10}), we can verify that
\begin{eqnarray}
W_{n+1}=(1+p)W_n+(1-p)W_{n-1}-W_{n-2}, \quad n\ge 2.
\label{ad-2.11}
\end{eqnarray}
We first show that $\{U_n^{[1]}\}_{n\ge 1}$ is log-concave. Let
$x_n=U^{[1]}_{n+1}/U_n^{[1]}$ for $n\ge 1$. It follows from
(\ref{ad-2.11}) that
\begin{eqnarray*}
x_n=1+p+\frac{1-p}{x_{n-1}}-\frac{1}{x_{n-1}x_{n-2}}, \quad n\ge 3.
\end{eqnarray*}
Then we have
\begin{eqnarray}
x_{n+1}-x_n=\frac{(p-1)(x_n-x_{n-1})}{x_{n-1}x_n}+\frac{x_n-x_{n-2}}{x_{n-2}x_{n-1}x_n},
\quad n\ge 3. \label{ad-2.12}
\end{eqnarray}
Now we prove by induction that $\{x_n\}_{n\ge 1}$ is decreasing. It
is clear that
\begin{eqnarray*}
&amp;&amp; U^{[1]}_1=1, \quad U_2^{[1]}=p+1, \quad U_3^{[1]}=p^2+p+2, \quad U_4^{[1]}=p^3+p^2+3p+2, \\
&amp;&amp;x_1=p+1\ge x_2=p+\frac{2}{p+1}&gt;x_3=p+\frac{p+2}{p^2+p+2}.
\end{eqnarray*}
Assume that $x_{k-1}\ge x_k$ for all $2\leq k\leq n$. It follows
from this assumption and (\ref{ad-2.12}) that $x_n\ge x_{n+1}$ for
$n\ge 1$. Hence $\{x_n\}_{n\ge 1}$ is decreasing, and
$\{U_n^{[1]}\}_{n\ge 1}$ is log-concave. Using this method, we can
also prove that $\{V_n^{[1]}\}_{n\ge 3}$ is log-concave. It follows
from Lemma \ref{lemma:1a} that $\{U_n^{[r]}\}_{n\geq 1}$ is
log-concave for $r\ge 2$.
\end{proof}
\begin{theorem} For $p\ge 1$, the sequences $\{n!U^{[1]}_n\}_{n\ge 5}$ and $\{n!V^{[1]}_n\}_{n\ge 5}$ are log-balanced.
\end{theorem}
\begin{proof} By (\ref{ad-2.9}), we derive
\begin{eqnarray}
U_n^{[1]}=\frac{U_n+U_{n+1}-1}{p}, \quad
V_n^{[1]}=\frac{V_n+V_{n+1}+p-2}{p}. \label{ad-2.13}
\end{eqnarray}
By (\ref{ad-2.13}) and (\ref{ad-2.10}), we can verify that
\begin{eqnarray*}
n\bigg(U_n^{[1]}\bigg)^2-(n+1)U_{n-1}^{[1]}U_{n+1}^{[1]}&amp;=&amp;\frac{(n+1)[(-1)^{n+1}+U_{n+1}-U_n]}{p}-\frac{(U_n+U_{n+1}-1)U_n^{[1]}}{p},\\
n\bigg(V_n^{[1]}\bigg)^2-(n+1)V_{n-1}^{[1]}V_{n+1}^{[1]}&amp;=&amp;\frac{(n+1)[(-1)^n\Delta-(p-2)(V_{n+1}-V_n)]}{p}\\
&amp;&amp;-\frac{(V_{n+1}+V_n+p-2)V^{[1]}_n}{p}.
\end{eqnarray*}
We can prove that
\begin{eqnarray*}
&amp;&amp; U_n^{[1]}\ge n+1, \quad n\ge 4,\\
&amp;&amp; V_n^{[1]}\ge n+1, \quad n\ge 4.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
&amp;&amp; U_n\ge 1, \quad n\ge 1,\\
&amp;&amp; V_n\ge p^2+2, \quad n\ge2.
\end{eqnarray*}
Then we get
\begin{eqnarray*}
n\bigg(U_n^{[1]}\bigg)^2-(n+1)U_{n-1}^{[1]}U_{n+1}^{[1]}&amp;\le&amp;\frac{2(n+1)(1-U_n)}{p}\\
&amp;\le&amp;0,\\
n\bigg(V_n^{[1]}\bigg)^2-(n+1)V_{n-1}^{[1]}V_{n+1}^{[1]}&amp;\le&amp;\frac{(n+1)[\Delta-(p-2)(V_{n+1}-V_n)-(V_{n+1}+V_n+p-2)]}{p}\\
&amp;=&amp;\frac{(n+1)[p^2-p+6-(p-1)(V_{n+1}-V_n)-2V_n]}{p}\\
&amp;\le&amp;\frac{(n+1)[-p^2-p+2-(p-1)(V_{n+1}-V_n)]}{p}\\
&amp;\le&amp;0.
\end{eqnarray*}
Then the sequences $\{n!U^{[1]}_n\}_{n\ge 5}$ and
$\{n!V^{[1]}_n\}_{n\ge 5}$ are log-convex for $p\ge 1$. Hence the
sequences $\{n!U^{[1]}_n\}_{n\ge 5}$ and $\{n!V_n^{[1]}\}_{n\ge 5}$
are log-balanced.
\end{proof}

\section{Conclusions}
We have discussed the log-concavity of hyperfibonacci numbers and
hyperlucas numbers. Our next work is to study the log-concavity
(log-convexity) of various recurrence sequences appearing in
combinatorics.

\section{Acknowledgment}
The authors are grateful to the anonymous referees for their
valuable comments and suggestions.

%==============================================================================================================
\begin{thebibliography}{99}
\bibliographystyle{plain}

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\bibitem{ref12} Y. Wang and Y.-N. Yeh,  Log-concavity and LC-positivity,
\textit{J. Combin. Theory Ser. A}. \textbf{114} (2007), 195--210.

\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}:  Primary 05A20;
Secondary 11B37, 11B39.

\noindent \emph{Keywords: } log-convexity, log-concavity, Fibonacci
number, Lucas number.

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\noindent (Concerned with sequences
\seqnum{A000071} and
\seqnum{A001610}.)

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\vspace*{+.1in}
\noindent
Received March 17 2013;
revised versions received  November 9 2013; November 24 2013.
Published in {\it Journal of Integer Sequences}, December 16 2013.

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\noindent
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\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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A000071
A001610
                                                                                

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