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\begin{center}
\vskip 1cm{\LARGE\bf 
Expansion of Fibonacci and Lucas 
\vskip .02in
Polynomials: An Answer to Prodinger's
\vskip .08in
Question
}
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\large

Hac\`{e}ne Belbachir\\
University of Sciences and Technology Houari Boumediene\\
Faculty of Mathematics\\
P. O. Box 32, El Alia \\
Bab-Ezzouar 16111, Algiers\\
Algeria \\
\href{mailto:hacenebelbachir@gmail.com}{\texttt{hacenebelbachir@gmail.com}} \\
\  \\
Athmane Benmezai\\
Faculty of Economy and Management Science\\
University of Dely Brahim\\
Rue Ahmed Ouaked\\
Dely Brahim, Algiers,\\
Algeria \\
\href{mailto:athmanebenmezai@gmail.com}{\texttt{athmanebenmezai@gmail.com}}
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\begin{abstract}
We give an answer to a recent question of Prodinger, which consists of
finding $q$-analogues of identities related to Fibonacci and Lucas
polynomials.
\end{abstract}

\section{Introduction}

The bivariate polynomials of Fibonacci and Lucas, denoted respectively by $%
\left( U_{n}\right) $\text{ and }$\left( V_{n}\right) $, are defined by%
\begin{equation*}
\left\{ 
\begin{array}{l}
U_{0}=0,\text{ }U_{1}=1, \\ 
U_{n}=tU_{n-1}+zU_{n-2}\text{ \ }\left( n\geq 2\right) ,%
\end{array}%
\right. \ \ \ \text{ and}\ \ \ \left\{ 
\begin{array}{l}
V_{0}=2,\text{ }V_{1}=t, \\ 
V_{n}=tV_{n-1}+zV_{n-2}\text{ }\left( n\geq 2\right) .%
\end{array}%
\right.
\end{equation*}

It is well-known (see, for example, \cite{bel1}) that%
\begin{equation*}
U_{n+1}=\sum_{k=0}^{\left\lfloor n/2\right\rfloor }{\binom{n-k}{k}}%
t^{n-2k}z^{k},\ \ \ \ \ \ \ V_{n}=\sum_{k=0}^{\left\lfloor n/2\right\rfloor }%
\frac{n}{n-k}{\binom{n-k}{k}}t^{n-2k}z^{k}\text{ }\left( n\geq 1\right) .
\end{equation*}

In \cite{zsx}, Belbachir and Bencherif proved the following five
formulae:

\begin{align}
2U_{2n+1}&amp; =\sum_{k=0}^{n}a_{n,k}t^{k}V_{2n-k}\text{,\ \ with \ }%
a_{n,k}=2\sum_{j=0}^{n}\left( -1\right) ^{j+k}\binom{j}{k}-\left( -1\right)
^{n+k}\binom{n}{k},  \label{aa} \\
V_{2n}&amp; =\sum_{k=1}^{n}b_{n,k}t^{k}V_{2n-k}\text{, with \ }b_{n,k}=\left(
-1\right) ^{k+1}\binom{n}{k},  \label{bb} \\
V_{2n-1}&amp; =\sum_{k=1}^{n}c_{n,k}t^{k}U_{2n-k}\text{, \ with \ }%
c_{n,k}=2\left( -1\right) ^{k+1}\binom{n}{k}-\left[ k=1\right] ,  \label{cc}
\\
2V_{2n-1}&amp; =\sum_{k=1}^{n}d_{n,k}t^{k}V_{2n-1-k}\text{,\ \ with\ \ }%
d_{n,k}=\left( -1\right) ^{k+1}\frac{2n-k}{n}\binom{n}{k},  \label{dd} \\
2U_{2n}&amp; =\sum_{k=1}^{n}e_{n,k}t^{k}V_{2n-k},\text{ \ with \ }  \label{ee} \\
e_{n,k}&amp; =\left( -1\right) ^{k+1}\frac{2n-k}{n}\binom{n}{k}%
+\sum_{j=0}^{n-1}\left( -1\right) ^{j+k-1}\binom{j}{k-1}-\frac{1}{2}\left(
-1\right) ^{n+k}\binom{n-1}{k-1}  \notag
\end{align}%
and they also gave the following simple sixth one:
\begin{equation}
V_{2n}=2U_{2n+1}-xU_{2n}.  \label{ff}
\end{equation}

The results presented in \cite{zsx} are companion generalizations for
Chebyshev polynomials \cite{bel2}.

Without loss of generality, we suppose that $t=1$.   We refer here to the
modified polynomials given by Prodinger in the introduction of \cite{rfv}.

In order to give $q$-analogues of these instances,\ Prodinger considered 
Cigler's \cite{edc}\ suggestion to replace $U_{n}$ and $V_{n}$, respectively,
with
\begin{equation*}
\mathbf{F}_{n+1}\left( z\right) =\sum_{k=0}^{\left\lfloor n/2\right\rfloor
}q^{\binom{k+1}{2}}{\QATOPD[ ] {n-k}{k}}_{q}z^{k}\text{,\ \ \ \ \ }\mathbf{L}%
_{n}\left( z\right) =\sum_{k=0}^{\left\lfloor n/2\right\rfloor }q^{\binom{k}{%
2}}{\QATOPD[ ] {n-k}{k}}_{q}\frac{[n]_{q}}{[n-k]_{q}}z^{k},
\end{equation*}%
with the following $q$-notations%
\begin{equation*}
\lbrack n]_{q}=1+q+\cdots +q^{n-1}\text{, \ \ }\left[ n\right]
_{q}!=[1]_{q}[2]_{q}\cdots \lbrack n]_{q}\text{,\ \ \ }{\QATOPD[ ] {n}{k}}%
_{q}=\frac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}.
\end{equation*}

Prodinger~\cite{rfv} found the following two $q$-analogues for
identities (\ref{dd}) and (\ref{ff}) respectively 
\begin{align}
\mathbf{F}_{2n}\left( z\right) &amp; =\sum_{k=1}^{n}\beta _{n,k}\mathbf{F}%
_{2n-k}\left( z\right) \text{, \ with \ }\beta _{n,k}=\left( -1\right)
^{k+1}q^{\binom{k}{2}}{\QATOPD[ ] {n}{k}}_{q},  \label{gg} \\
\mathbf{L}_{2n-1}&amp; =\sum_{k=1}^{n}\delta _{n,k}\mathbf{L}_{2n-1-k}\left(
z\right) ,\ \ \text{with \ }\delta _{n,k}=\left( -1\right) ^{k+1}\frac{q^{%
\binom{k}{2}}}{1+q^{n-1}}\left( {\QATOPD[ ] {n-1}{k}}_{q}+q^{n-1}{\QATOPD[ ]
{n}{k}}_{q}\right) ,  \label{hh}
\end{align}%
and left the three others as a challenge.



We give an answer to the remaining three instances and we\ find a $q$%
-analogue for identity (\ref{ff}).

\section{An answer to Prodinger's question}

Here, we find a $q$-analogue for the identities (\ref{ff}), (\ref{cc}), (\ref%
{aa}) and (\ref{ee}), which can be considered as an answer to Prodinger's
question \cite{rfv}.

\begin{theorem}
The following $q$-identities hold%
\begin{align}
\mathbf{L}_{2n}\left( z\right) &amp; =\mathbf{F}_{2n+1}\left( \frac{z}{q}\right)
+\mathbf{F}_{2n+1}\left( z\right) -\mathbf{F}_{2n}\left( z\right) ,
\label{ii} \\
\mathbf{L}_{2n-1}\left( z\right) &amp; =\sum_{k=1}^{n}\beta _{n,k}\left( \mathbf{%
F}_{2n-k}\left( z\right) +\mathbf{F}_{2n-k}\left( \frac{z}{q}\right) \right)
-\mathbf{F}_{2n-1}\left( z\right) ,  \label{kk} \\
\mathbf{F}_{2n+1}\left( z\right) &amp; =\frac{1}{1-q^{2n}}\left( \mathbf{L}%
_{2n}\left( qz\right) -q^{2n}\mathbf{L}_{2n}\left( z\right) \right) ,
\label{ll} \\
\mathbf{F}_{2n}\left( z\right) &amp; =\frac{1}{1-q^{2n-1}}\left( \mathbf{L}%
_{2n-1}\left( qz\right) -q^{2n-1}\mathbf{L}_{2n-1}\left( z\right) \right) .
\label{mm}
\end{align}
\end{theorem}

\begin{proof}
We have%
\begin{align*}
\mathbf{F}_{n+1}\left( \frac{z}{q}\right) +\mathbf{F}_{n+1}\left( z\right) -%
\mathbf{F}_{n}\left( z\right) &amp; =\sum\nolimits_{k=0}^{\left\lfloor
n/2\right\rfloor }q^{\binom{k}{2}}{\QATOPD[ ] {n-k}{k}}_{q}\left(
1+q^{k}-q^{k}\frac{\left[ n-2k\right] _{q}}{\left[ n-k\right] _{q}}\right)
z^{k} \\
&amp; =\sum\nolimits_{k=0}^{\left\lfloor n/2\right\rfloor }q^{\binom{k}{2}}{%
\QATOPD[ ] {n-k}{k}}_{q}\left( 1+q^{k}-q^{k}\frac{1-q^{n-2k}}{1-q^{n-k}}%
\right) z^{k} \\
&amp; =\sum\nolimits_{k=0}^{\left\lfloor n/2\right\rfloor }q^{\binom{k}{2}}{%
\QATOPD[ ] {n-k}{k}}_{q}\frac{1-q^{n}}{1-q^{n-k}}z^{k} \\
&amp; =\sum\nolimits_{k=0}^{\left\lfloor n/2\right\rfloor }q^{\binom{k}{2}}{%
\QATOPD[ ] {n-k}{k}}_{q}\frac{[n]_{q}}{[n-k]_{q}}z^{k}=\mathbf{L}_{n}\left(
z\right) .
\end{align*}%
For the second identity $\sum_{k=1}^{n}\beta _{n,k}\left( \mathbf{F}%
_{2n-k}\left( z\right) +\mathbf{F}_{2n-k}\left( \frac{z}{q}\right) \right) -%
\mathbf{F}_{2n-1}\left( z\right) =\mathbf{F}_{2n}\left( z\right) +\mathbf{F}%
_{2n}\left( \frac{z}{q}\right) -\mathbf{F}_{2n-1}\left( z\right) =\mathbf{L}%
_{2n-1}\left( z\right) .$

To establish (\ref{ll}) and (\ref{mm}), we have
\begin{eqnarray*}
\mathbf{L}_{n}\left( qz\right) -q^{n}\mathbf{L}_{n}\left( z\right)
&amp;=&amp;\sum\nolimits_{k=0}^{\left\lfloor n/2\right\rfloor }q^{\binom{k}{2}}{%
\QATOPD[ ] {n-k}{k}}_{q}\frac{\left[ n\right] _{q}}{\left[ n-k\right] _{q}}%
\left( q^{k}-q^{n}\right) z^{k} \\
&amp;=&amp;\sum\nolimits_{k=0}^{\left\lfloor n/2\right\rfloor }q^{\binom{k+1}{2}}{%
\QATOPD[ ] {n-k}{k}}_{q}\left( 1-q^{n}\right) z^{k} \\
&amp;=&amp;\left( 1-q^{n}\right) \mathbf{F}_{n+1}\left( z\right) ,
\end{eqnarray*}%
and then take $n$ odd and $n$ even respectively.
\end{proof}

\begin{remark}
We have to notice that the expressions of relations (\ref{ll}) and (\ref{mm}%
) do not give, for $q=1$, the initial formulae as given in relations (\ref%
{aa}) and (\ref{ee}). We obtain, when $q\rightarrow 1,$ the well known
identity $U_{n+1}\left( z\right) =V_{n}\left( z\right) -zU_{n-1}\left(
z\right) ,$ for $n$ odd and $n$ even respectively. However, they express, as
in Belbachir's and Bencherif's work \cite{zsx}, an odd Fibonacci polynomial
in terms of Lucas polynomials starting with a smallest even index, and an
even Fibonacci polynomial in terms of Lucas polynomials starting with a
smallest odd index.
\end{remark}

\section{Acknowledgment}

The authors would like to thank the referee for his patience and very
useful guidance which improved the quality of the paper.

\begin{thebibliography}{9}
\bibitem{zsx} H. Belbachir and F. Bencherif, On some properties of bivariate
Fibonacci and Lucas polynomials, \textit{J. Integer Sequences }\textbf{11}
(2008), \href{http://cs.uwaterloo.ca/journals/JIS/VOL11/Belbachir/belbachir13.html}{Article 08.2.6}.

\bibitem{bel2} H. Belbachir and F. Bencherif, On some properties of
Chebyshev polynomials, \textit{Discuss. Math. Gen. Algebra Appl} \textbf{28}
(2008), 121--133.

\bibitem{bel1} H. Belbachir and F. Bencherif, Linear recurrent sequences and
powers of a square matrix. \textit{Integers} \textbf{6} (2006), Paper A12.

\bibitem{edc} J. Cigler, A new class of $q$-Fibonacci polynomials, \textit{%
Electronic J. Combinatorics} \textbf{10} (2003), 
\href{http://www.combinatorics.org/ojs/index.php/eljc/article/view/v10i1r19}{Article R19}.

\bibitem{rfv} H. Prodinger, On the expansion of Fibonacci and Lucas
polynomials,\textit{\ J. Integer Sequences }\textbf{12} (2009), 
\href{https://cs.uwaterloo.ca/journals/JIS/VOL12/Prodinger/prodinger27.html}{Article 09.1.6}.
\end{thebibliography}

\bigskip
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\noindent 2000 \textit{Mathematics Subject Classification}: Primary 11B39;
Secondary 05A30, 11B37.

\noindent \textit{Keywords}: Fibonacci polynomials; Lucas polynomials; $q$%
-analogues.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A007318},
\seqnum{A029653}, and
\seqnum{A112468}.)

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received July 15 2012;
revised versions received  August 14 2012; September 3 2012.
Published in {\it Journal of Integer Sequences}, September 8 2012.

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