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\begin{center}
\vskip 1cm{\LARGE\bf On Generalized Fibonacci Polynomials and
Bernoulli Numbers\footnote{
This work is supported by the N. S. F. (10271093, 60472068) and
P. N. S. F. of P. R. China.}
}
\vskip 1cm
\large
Tianping Zhang\footnote{Author's current address:
College of Mathematics and Information Science,
Shannxi Normal University,
Xi'an, Shaanxi, P. R. China.}\\
Department of Mathematics \\
Northwest University\\
Xi'an, Shaanxi\\
P. R. China\\
\href{mailto:tpzhang@snnu.edu.cn}{\tt tpzhang@snnu.edu.cn} \\
\bigskip
Yuankui Ma \\
Department of Mathematics and Physics \\
Xi'an Institute of Technology\\
Xi'an, Shaanxi \\
P. R. China\\
\href{mailto:ykma@eyou.com}{\tt ykma@eyou.com} \\
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\vskip .2 in
\begin{abstract}
In this paper we use elementary methods to study the
relationship between the generalized Fibonacci polynomials and the
famous Bernoulli numbers, and give several interesting identities
involving them.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{lemma}{Lemma}

\section{Introduction and results}

As usual, the famous Fibonacci polynomials $F(x) = \{F_n(x)\}$ are
defined by the second-order linear recurrence 
$$
F_{n+2}(x) = x F_{n+1}(x) + F_n(x) \eqno{(1)}
$$
for $n \ge 0$ and $F_0(x) = 0$, $F_1(x) = 1$. These
polynomials are of great importance in the study of many subjects
such as algebra, geometry, and number theory itself. Obviously, they have
a deep relationship with the famous Fibonacci numbers $(F_n)_{n \geq 0}$.
That is, $F_n(1)=F_n$. Many scholars have studied numerous properties
of the Fibonacci numbers. For example, R. L. Duncan \cite{1} and
L. Kuipers \cite{2} proved that $(\log F_n)$ is uniformly
distributed mod $1$. N. Robbins \cite{3} studied the Fibonacci
numbers of the forms $px^2\pm 1$, $px^3 \pm 1$, where $p$  is a
prime. Wenpeng Zhang \cite{4} and Fengzhen Zhao \cite{5} obtained
some identities involving the Fibonacci numbers. Moreover, Yuan Yi
and Wenpeng Zhang \cite{6} studied the calculation on the
summation involving the Fibonacci polynomials, and obtained the
following

\begin{proposition}
Let $F(x) = \{F_n(x)\}$ be defined by (1).
Then for all positive integers $k$ and $n$, we have the
formula
$$
\sum_{a_1 + a_2 + \dots + a_k = n} F_{a_1+1}(x) F_{a_2+1}(x)
\cdots F_{a_k+1}(x) =
\sum_{m=0}^{\lfloor \frac{n}{2} \rfloor}{{n+k-1-m} \choose m}
{{n+k-1-2m} \choose {k-1}} x^{n-2m}
$$
where the summation is over all $k$-dimension nonnegative integer
coordinates $(a_1$, $a_2$, $\dots$, $a_k)$ such that $a_1 + a_2 +
\dots + a_k = n$, and $\lfloor z \rfloor$ denotes the greatest integer not
exceeding $z$, and $ {m \choose n} =
\frac{m!}{n!(m-n)!}$.
\end{proposition}

On the other hand, the famous Bernoulli numbers are defined by
$$
\frac{x}{e^x-1}=\sum_{n=0}^\infty B_{n}\frac{x^{n}}{n!},\quad \mid
x \mid<2\pi. \eqno{(2)}
$$
A recursion formula involving the Bernoulli numbers is
$$
B_n=\sum_{k=0}^n {n \choose k}B_k
$$
for $n\ge 2$ and $B_0=1, B_1=-\frac{1}{2}$, which successively yields
the values
$$
B_2=\frac{1}{6},\quad B_{2k+1}=0,(k=1,2,\cdots), \quad 
B_4=-\frac{1}{30},\quad B_6=\frac{1}{42},\\
$$
$$
B_8=-\frac{1}{2}, \quad B_{10}=\frac{5}{66}, \quad
B_{12}=-\frac{691}{2730}, \quad B_{14}=\frac{7}{6}, \quad\cdots
$$
Moreover, the Bernoulli numbers $B_{2k}$ alternate in sign, and
are related to $\zeta(2k)$ as follows:
$$
\zeta(2k)=(-1)^{k+1}\frac{(2\pi)^{2k}B_{2k}}{2(2k)!}.
$$
Other important results involving the Bernoulli numbers can be
found in references \cite{7,8,9}. 

Now we consider the polynomial sequence
$H(x)=\{H_n(x)\}$ defined by $H_0(x) = 0$, $H_1(x) = 1$, and
$$
H_{n+2}(x) = P(x) H_{n+1}(x) + Q(x) H_n(x), \eqno{(3)}
$$
where $P(x)$ and $Q(x)$ are polynomials with
$\Delta(x)=P^2(x)+4Q(x)>0$. It is easy to see that $(3)$ is a
generalization of $(1)$.

It is well known that the Fibonacci numbers and Lucas numbers
are closely related to the Chebyshev polynomials. 
Yuankui Ma and the first author
\cite{10} studied the relationships between the
Chebyshev polynomials of the first kind and the famous Euler numbers, and
obtained an interesting identity involving them.  But no similar
relationship 
between the generalized Fibonacci polynomials and
the Bernoulli numbers was previously known.
In this paper,
we use elementary methods to study the relationship
between the generalized Fibonacci polynomials and the famous
Bernoulli numbers, and give several interesting identities
involving them. That is, we shall prove the following

\begin{theorem}
For all positive integers $k$ and $n$
with $k\le n$, we have the formula
$$
\sum_{a_1+\cdots+a_k+b_1+\cdots+b_k=n}\frac{H_{a_1}(x)}{a_1!}
\cdots \frac{H_{a_k}(x)}{a_k!} \frac{B_{b_1}}{b_1!} \cdots
\frac{B_{b_k}}{b_k!}
\left(\sqrt{\Delta(x)}\right)^{b_1+\cdots+b_k}=\frac{(k\beta)^{n-k}
}{(n-k)!},
$$
where $\beta=\frac{P(x)-\sqrt{\Delta(x)}}{2}.$
\end{theorem}

If we take $P(x)=x$ and $Q(x)=1$ in Theorem 1, then we have

\begin{corollary} % 1
For all positive integers $k$ and $n$
with $k\le n$, we have
$$
\sum_{a_1+\cdots+a_k+b_1+\cdots+b_k=n}\frac{F_{a_1}(x)}{a_1!}
\cdots \frac{F_{a_k}(x)}{a_k!} \frac{B_{b_1}}{b_1!} \cdots
\frac{B_{b_k}}{b_k!}
\left(\sqrt{x^2+4}\right)^{b_1+\cdots+b_k}=\frac{(k\beta(x))^{n-k}
}{(n-k)!},
$$
where $\beta(x)=\frac{x-\sqrt{x^2+4}}{2}.$
\end{corollary}

If $P(x)$ and nonzero $Q(x)$ in Theorem 1 are integers
with $P^2+4Q>0,$  then we immediately obtain the following

\begin{corollary} % 2
For all positive integers $k$ and $n$
with $k\le n$, we have the identity
$$
\sum_{a_1+\cdots+a_k+b_1+\cdots+b_k=n}\frac{H_{a_1}}{a_1!} \cdots
\frac{H_{a_k}}{a_k!} \frac{B_{b_1}}{b_1!} \cdots
\frac{B_{b_k}}{b_k!}
\left(\sqrt{P^2+4Q}\right)^{b_1+\cdots+b_k}=\frac{(k\beta ')^{n-k}
}{(n-k)!},
$$
where $\beta '=\frac{P-\sqrt{P^2+4Q}}{2}.$
\end{corollary}

Taking $x=1$ in Corollary 1, or $P=Q=1$ in Corollary 2, we 
immediately deduce the following

\begin{corollary} % 3
For all positive integers $k$ and $n$
with $k\le n$, we have the identity
$$
\sum_{a_1+\cdots+a_k+b_1+\cdots+b_k=n}\frac{F_{a_1}}{a_1!} \cdots
\frac{F_{a_k}}{a_k!} \frac{B_{b_1}}{b_1!} \cdots
\frac{B_{b_k}}{b_k!}
\left(\sqrt{5}\right)^{b_1+\cdots+b_k}=\frac{(k\beta(1))^{n-k}
}{(n-k)!},
$$
where $\beta(1)=\frac{1-\sqrt{5}}{2}.$
\end{corollary}

In particular, taking $k=1,2,3$ in Corollary 3, we easily get

\begin{corollary} %4
For all positive integers $n$, we have
$$
\sum_{a+b=n}\frac{F_{a}}{a!}\frac{B_{b}}{b!}
\left(\sqrt{5}\right)^{b}=\frac{(\beta(1))^{n-1} }{(n-1)!}.
$$
\end{corollary}

\begin{corollary} %5
For all positive integers $n\ge 2$, we
have
$$
\sum_{a+b+c+d=n}\frac{F_{a}}{a!}  \frac{F_b}{b!} \frac{B_{c}}{c!}
\frac{B_d}{d!} \left(\sqrt{5}\right)^{c+d}=\frac{(2\beta(1))^{n-2}
}{(n-2)!}.
$$
\end{corollary}

\begin{corollary} %6
For all positive integers $n\ge 3$,
we have
$$
\sum_{a+b+c+d+e+f=n}\frac{F_{a}}{a!} \frac{F_b}{b!}\frac{F_c}{c!}
\frac{B_{d}}{d!} \frac{B_e}{e!}\frac{B_f}{f!}
\left(\sqrt{5}\right)^{d+e+f}=\frac{(3\beta(1))^{n-3} }{(n-3)!}.
$$
\end{corollary}

\section{Proof of Theorem}

In this section, we shall complete the proof of Theorem. First we
let $\alpha=\frac{P(x)+\sqrt{\Delta(x)}}{2}$ and
$\beta=\frac{P(x)+\sqrt{\Delta(x)}}{2}$ denote the roots of
characteristic polynomial $\lambda^2-P(x)\lambda-Q(x)$ of the
generalized Fibonacci polynomial sequence $H(x)$, then the terms
of the sequence $H(x)$ can be expressed as (see \cite{11,12})
$$
\allowdisplaybreaks\displaystyle H_n(x) =
\frac{1}{\sqrt{\Delta(x)}}\left(\left(\frac{P(x)+\sqrt{\Delta(x)}}{2}\right)^n
- \left(\frac{P(x) - \sqrt{\Delta(x)}}{2}\right)^n \right).
$$
Then we easily deduce that the generating function of $H(t,x)$
is
$$
\allowdisplaybreaks\displaystyle H(t,x)=\sum_{n=0}^\infty
\frac{H_n(x)}{n!}t^n=\sum_{n=0}^\infty\frac{\alpha^n
-\beta^n}{(\alpha-\beta)n!}t^n. \eqno{(4)}
$$

That is,
$$
\allowdisplaybreaks\displaystyle H(t,x)=\frac{e^{\alpha
t}-e^{\beta t}}{\alpha-\beta} =\frac{e^{\beta
t}\left(e^{t\sqrt{\Delta(x)}}-1\right)}{\sqrt{\Delta(x)}}.
$$
Therefore, we have
$$
\allowdisplaybreaks\displaystyle e^{\beta t}=
\frac{H(t,x)}{t}\cdot\frac{t\sqrt{\Delta(x)}}{e^{t\sqrt{\Delta(x)}}-1}.
$$
Then from (2) and (4), we have
$$
\allowdisplaybreaks\displaystyle e^{\beta t}=
\left(\sum_{m=0}^\infty \frac{H_m(x)}{m!}t^{m-1}\right)
\left(\sum_{n=0}^\infty
\frac{B_n}{n!}\left(t\sqrt{\Delta(x)}\right)^n\right). \eqno{(5)}
$$

Note that for two absolutely convergent power series
$\displaystyle\sum_{n=0}^{\infty}a_n t^n$ and
$\displaystyle\sum_{n=0}^{\infty}b_n t^n$, we have
$$
\allowdisplaybreaks\left(\sum_{n=0}^{\infty}a_n t^n\right)\cdot
\left(\displaystyle\sum_{n=0}^{\infty}b_n t^n\right)
=\sum_{n=0}^{\infty}\left(\sum_{u+v=n}a_u b_v\right)t^n,
$$
so $k$ times on the both sides of formula (5), we have
$$\allowdisplaybreaks
LHS= \left(e^{\beta t}\right)^k = e^{k\beta t}= \sum_{n=0}^\infty \frac{\left(k\beta\right)^n}{n!}t^n,
$$
$$
\allowdisplaybreaks\displaystyle RHS= \sum_{n=0}^\infty
\sum_{a_1+\cdots+a_k+b_1+\cdots+b_k=n}\frac{H_{a_1}(x)}{a_1!}
\cdots \frac{H_{a_k}(x)}{a_k!} \frac{B_{b_1}}{b_1!} \cdots
\frac{B_{b_k}}{b_k!}
\left(\sqrt{\Delta(x)}\right)^{b_1+\cdots+b_k}t^{n-k} .
$$

Comparing the coefficients of $t^{n-k}$ on the above, we 
immediately obtain the following identity
$$
\allowdisplaybreaks
\displaystyle\sum_{a_1+\cdots+a_k+b_1+\cdots+b_k=n}\frac{H_{a_1}(x)}{a_1!}
\cdots \frac{H_{a_k}(x)}{a_k!} \frac{B_{b_1}}{b_1!} \cdots
\frac{B_{b_k}}{b_k!}
\left(\sqrt{\Delta(x)}\right)^{b_1+\cdots+b_k}=\frac{(k\beta)^{n-k}
}{(n-k)!}.
$$

This completes the proof of Theorem 1.

\section{Acknowledgments}


The author wishes to thank his supervisor, Professor
Wenpeng Zhang, who has been most generous with his advice and
support. Moreover, the author also thanks the anonymous referee
for his very helpful and detailed comments on the original
manuscripts.

\begin{thebibliography}{10}
\bibliographystyle{plain}

\bibitem{1} R. L. Duncan, Application of uniform
distribution to the Fibonacci numbers,
{\it Fibonacci Quart.} {\bf 5} (1967), 137--140.

\bibitem{2}
L. Kuipers,  Remark on a paper by R. L. Duncan
concerning the uniform distrubution mod 1 of the sequence of the
logarithms of the Fibonacci numbers,
{\it Fibonacci Quart.} {\bf 7} (1969), 465--466.

\bibitem{3}
N. Robbins,
Fibonacci numbers of the forms $px^2 \pm 1, px^3 \pm 1$,
where $p$ is prime, in 
{\it Applications of Fibonacci Numbers},
Kluwer Academic, 1986, pp.\ 77--88.

\bibitem{4}
Wenpeng Zhang, Some identities involving the
Fibonacci numbers, {\it Fibonacci Quart.} {\bf 35} (1997),
225--229.

\bibitem{5}
Fengzhen Zhao and Tianming Wang, 
Generalizations of some identities involving the Fibonacci
polynomials, {\it Fibonacci Quart.} {\bf 39} (2001),
165--167.


\bibitem{6}
Yuan Yi and Wenpeng Zhang,  Some identities
involving the Fibonacci polynomials,
{\it Fibonacci Quart.}
{\bf 40} (2002), 314--318.


\bibitem{7}
Gi-Sang Cheon, A note on the Bernoulli and
Euler polynomials, {\it Appl. Math. Lett.} {\bf 16}
(2003), 365--368.

\bibitem{8} 
H. M. Srivastava and \'A. Pint\'er,
Remarks on some relationships between the Bernoulli and
Euler polynomials,
{\it Appl. Math. Lett.} {\bf 17}
(2004), 375--380.

\bibitem{9}
M. Apostol, {\it Introduction to Analytic Number
Theory}, Springer-Verlag, New York, 1976.

\bibitem{10} Yuankui Ma and Tianping Zhang,
An identity
involving the first-kind Chebyshev polynomials and the Euler
numbers, {\it J. Ningxia University}, to appear.

\bibitem{11}
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\bibitem{12}
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\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11B37; Secondary 11B39.

\noindent \emph{Keywords: } 
generalized Fibonacci polynomials; Bernoulli numbers;

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequence
\seqnum{A000045}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received July 14 2005;
revised version received  October 14 2005.
Published in {\it Journal of Integer Sequences}, October 21 2005.

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