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\begin{center}
\vskip 1cm{\LARGE\bf Relating Balancing Polynomials to  \\
Lucas-Balancing  Polynomials \\
\vskip .08in
via Bernoulli Numbers}
\vskip 1cm
\large
Mouloud Goubi\\
Department of Mathematics\\
UMMTO University \\
Laboratory of Algebra and Number Theory (USTHB)\\
Tizi Ouzou\\
Algeria\\
\href{mailto:mouloud.goubi@ummto.dz}{\tt mouloud.goubi@ummto.dz}\\
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\vskip .2 in

\begin{abstract}
We derive relations for balancing and Lucas-balancing
polynomials. The results provide extensions of some results proved
recently by Frontczak.
\end{abstract}

\section{Introduction}
The Fibonacci numbers $F_n$ and Lucas numbers $L_n$ \cite{Byrd,VERN}
are sequences satisfying the Fibonacci recursion relation
\begin{equation}
X_{n+1}=X_n+X_{n-1}.
\end{equation}
The initial conditions are, respectively, $F_0=0,\ F_1=1$ and $L_0=2,\
L_1=1$. The first elements of the sequences are 
$$F: 0, 1, 1, 2, 3, 5, 8, 13, 21,\cdots$$
and 
$$L : 2, 1, 3, 4, 7, 11, 18, 29, 47,\cdots .$$

Fibonacci numbers $\left(F_n\right)_{n\geq0}$ and Lucas numbers
$\left(L_n\right)_{n\geq0}$ are regarded as the some of the most important
sequences in mathematics. Relationships between them have been
studied in the past. Hoggatt \cite{VERN} states as exercises several
identities relating Fibonacci numbers with Lucas numbers. 

These
numbers are also related to balancing and Lucas-balancing
polynomials. Balancing polynomials $B^{*}_{n}(x)$ are defined by
the recurrence (see \cite{Frontczak2, Frontczak3, Patel})
\begin{equation}
B^{*}_{n}(x)=6xB^{*}_{n-1}(x)-B^{*}_{n-2}(x), \ n\geq2,
\end{equation}
with initial terms $B^{*}_{0}(x)=0$ and
$B^{*}_{1}(x)=1$. Lucas-balancing polynomials $C_{n}(x)$ are
defined by
\begin{equation}
C_{n}(x)=6xC_{n-1}(x)-C_{n-2}(x), \ n\geq2,
\end{equation}
with initial terms $C_0(x)=1$ and $C_1(x)=3x$. The numbers
$B^{*}_{n}=B^{*}_{n}(1)$ (or $C_{n}=C_{n}(1)$) are so-called
balancing numbers (or Lucas-balancing numbers). The Binet formulas
for $B^{*}_{n}(x)$ and $C_{n}(x)$ are
\begin{equation}
B^{*}_{n}(x)=\frac{\lambda^n(x)-\lambda^{-n}(x)}{\lambda(x)-\lambda^{-1}(x)}\
\textrm{and}\
C_n(x)=\frac{1}{2}\left(\lambda^n(x)+\lambda^{-n}(x)\right),
\end{equation}
where $\lambda(x)=3x+\sqrt{9x^2-1}$ and 
$\lambda^{-1}(x)=3x-\sqrt{9x^2-1}$. We refer to Frontczak
\cite{Frontczak2} for more details. Accordingly, the exponential
generating functions of polynomials $B^{*}_n(x)$ and $C_n(x)$
are given by Frontczak \cite[Lemma 2, p.~3]{Frontczak} as follows:
\begin{equation}
F\left(x,t\right)=\frac{e^{3xt}}{\sqrt{9x^2-1}}\sinh\left(t\sqrt{9x^2-1}\right)=\sum_{n\geq0}B^{*}_n(x)\frac{t^n}{n!},
\end{equation}
and
\begin{equation}
G\left(x,t\right)=e^{3xt}\cosh\left(t\sqrt{9x^2-1}\right)=\sum_{n\geq0}C_n(x)\frac{t^n}{n!},
\end{equation}
where $\sinh$ and $\cosh$ are the hyperbolic sine and cosine
functions. The relations that we spoke about previously are given
in the papers \cite{Frontczak, Frontczak2} as follows:
\begin{equation}\label{bn}
B^{*}_n\left(\frac{L_{2m}}{6}\right)=\frac{F_{2mn}}{F_{2m}},\
C_n\left(\frac{L_{2m}}{6}\right)=\frac{L_{2m}}{2},
\end{equation}
 and
 \begin{equation}\label{cn}
B^{*}_n\left(\frac{i}{6}L_{2m+1}\right)=i^{n-1}\frac{F_{(2m+1)n}}{F_{2m+1}},\
C_n\left(\frac{i}{6}L_{2m+1}\right)=i^n\frac{L_{(2m+1)n}}{2},
\end{equation}
where $m$ is an integer and $i=\sqrt{-1}$ is the imaginary unit.

The Cauchy product of exponential generating functions is
defined by
\begin{equation}
\left(\sum_{n\geq0}a_n(x)\frac{t^n}{n!}\right)\left(\sum_{n\geq0}b_n(x)\frac{t^n}{n!}\right)=\sum_{n\geq0}\left(\sum_{k=0}^{n}{n\choose
k}a_k(x)b_{n-k}(x)\right) \frac{t^n}{n!}.
\end{equation}
This product allows us to provide extensions of Theorem 3 and
Corollary 3 proved by Frontczak \cite{Frontczak} in 2019.

\section{Connection between balancing polynomials and Lucas-balancing polynomials}
As usual, we use $B_n$ for the $n^{\rm th}$ \emph{Bernoulli
number} and $E_n$ for the $n^{\rm th}$ \emph{Euler number}.
These classical numbers (see Apostol \cite{Apost}) are defined
respectively by
\begin{equation}
B(t)=\frac{t}{e^t-1}=\sum_{k\geq0}B_k\frac{t^k}{k!},\ |t|<2\pi
\end{equation}
and
\begin{equation}
E(t)=\frac{2}{e^t+1}=\sum_{k\geq0}E_k\frac{t^k}{k!},\ |t|<\pi.
\end{equation}
The first few Bernoulli numbers are $B_0=1$, $B_1=-\frac{1}{2}$,
$B_2=\frac{1}{6}$. Also $B_{2n+1}=0$ for $n\geq1$. The first few Euler
numbers are $E_0=1$, $E_2=-1$, $E_4=5$ and $E_{2n+1}=0$ for
$n\geq0$. 

Frontczak \cite[Theorem
3]{Frontczak} proved the following identities
\begin{equation}\label{eq1}
\sum_{{k=0}\atop {n\equiv k \;{\rm(mod\;2)}}}^{n} {n\choose
k}\left(2\sqrt{9x^2-1}\right)^{n-k}B_{n-k}B^{*}_k(x)=nC_{n-1}(x) 
\end{equation}
and
\begin{equation}\label{eq2}
\sum_{k=0\atop {n\equiv k \;{\rm(mod\;2)}}}^{n}{n\choose
k}\left(2\sqrt{9x^2-1}\right)^{n-k}\left(2^{n-k}-1\right)B_{n-k}C_k(x)=n\left(9x^2-1\right)B^{*}_{n-1}(x),
\end{equation}
where $n\equiv k$ (mod $2$) means that $n-k$ is a multiple of $2$. An
improvement of the identities \eqref{eq1} and \eqref{eq2} is given
in the following theorem:
\begin{theorem}\label{th1}
Let $n\geq1$. Then
\begin{equation}\label{eq1th1}
\begin{split}
\sum_{k=0}^{n}{n\choose
k}\left(2\sqrt{9x^2-1}\right)^{n-k}B_{n-k}B^{*}_k(x)=n\left(C_{n-1}(x)-\sqrt{9x^2-1}B^{*}_{n-1}(x)\right)
\end{split}
\end{equation}
and
\begin{equation}\label{eq2th2}
\begin{split}
\sum_{k=0}^{n}{n\choose
k}\left(2\sqrt{9x^2-1}\right)^{n-k}\left(2^{n-k}-1\right)B_{n-k}C_k(x)=n\left(9x^2-1\right)B^{*}_{n-1}(x)-n\sqrt{9x^2-1}C_{n-1}(x).
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
Let the polynomial
\begin{equation*}
A_n(x)=\sum_{k=0}^{n}{n\choose
k}B^{*}_{n-k}(x)\left(2\sqrt{9x^2-1}\right)^{k}B_{k}.
\end{equation*}
Then $A_n(x)$ is defined in means of the Cauchy product of the
functions
\begin{equation*}
\frac{e^{3xt}}{\sqrt{9x^2-1}}\sinh\left(t\sqrt{9x^2-1}\right)\
\textrm{and}\ \frac{2t\sqrt{9x^2-1}}{e^{2t\sqrt{9x^2-1}}-1}.
\end{equation*}
But the product is
\begin{equation*}
A(x,t)=\frac{2te^{3xt}}{e^{2t\sqrt{9x^2-1}}-1}\sinh\left(t\sqrt{9x^2-1}\right).
\end{equation*}
Since
\begin{equation*}
\coth
t=1+\frac{2}{e^{2t}-1}=1+\frac{1}{t}\sum_{n\geq0}2^nB_n\frac{t^n}{n!}
\end{equation*}
 then
 \begin{equation*}
A(x,t)=te^{3xt}\left(\coth\left(t\sqrt{9x^2-1}\right)-1\right)\sinh\left(t\sqrt{9x^2-1}\right).
\end{equation*}
Furthermore
\begin{equation*}
A(x,t)=te^{3xt}\cosh\left(t\sqrt{9x^2-1}\right)
-te^{3xt}\sinh\left(t\sqrt{9x^2-1}\right)
\end{equation*}
Finally
\begin{equation*}
A(x,t)=tG(x,t)-t\sqrt{9x^2-1}F(x,t)
\end{equation*}
and
\begin{equation*}
\frac{2te^{3xt}}{e^{2t\sqrt{9x^2-1}}-1}\sinh\left(t\sqrt{9x^2-1}\right)=\sum_{n\geq0}n\left(C_{n-1}(x)-\sqrt{9x^2-1}B^{*}_{n-1}(x)\right)\frac{t^n}{n!}.
\end{equation*}
Thus $A_n(x)$ is identical with the polynomial
\begin{equation*}
n\left(C_{n-1}(x)-\sqrt{9x^2-1}B^{*}_{n-1}(x)\right),
\end{equation*}
and the result \eqref{eq1th1} follows. Let the polynomial
$$D_n(x)=\sum_{k=0}^{n}{n\choose
k}C_k(x)\left(2\sqrt{9x^2-1}\right)^{n-k}\left(2^{n-k}-1\right)B_{n-k}.$$
Then $D_n(x)=J_n(x)-K_n(x)$ with
\begin{equation*}
J_n(x)=\sum_{k=0}^{n}{n\choose
k}C_k(x)\left(4\sqrt{9x^2-1}\right)^{n-k}B_{n-k}
\end{equation*}
and
\begin{equation*}
K_n(x)=\sum_{k=0}^{n}{n\choose
k}C_k(x)\left(2\sqrt{9x^2-1}\right)^{n-k}B_{n-k}.
\end{equation*}
But the polynomial $J_n(x)$ is generated by the function
\begin{equation*}
\frac{4t\sqrt{9x^2-1}}{e^{4t\sqrt{9x^2-1}}-1}e^{3xt}\cosh\left(t\sqrt{9x^2-1}\right)
\end{equation*}
and the polynomial $K_n(x)$ is generated by the function
\begin{equation*}
\frac{2t\sqrt{9x^2-1}}{e^{2t\sqrt{9x^2-1}}-1}e^{3xt}
\cosh\left(t\sqrt{9x^2-1}\right).
\end{equation*}
Furthermore the generating function of the polynomial $D_n(x)$ is
\begin{equation*}
D(x,t)=\frac{4t\sqrt{9x^2-1}}{e^{4t\sqrt{9x^2-1}}-1}e^{3xt}\cosh\left(t\sqrt{9x^2-1}\right)-\frac{2t\sqrt{9x^2-1}}{e^{2t\sqrt{9x^2-1}}-1}e^{3xt}
\cosh\left(t\sqrt{9x^2-1}\right).
\end{equation*}
Then
\begin{equation*}
D(x,t)=\frac{2t\sqrt{9x^2-1}}{e^{2t\sqrt{9x^2-1}}-1}\left(\frac{2}{e^{2t\sqrt{9x^2-1}}+1}-1\right)e^{3xt}\cosh\left(t\sqrt{9x^2-1}\right)
\end{equation*}
and
\begin{equation*}
D(x,t)=-\frac{2t\sqrt{9x^2-1}}{e^{2t\sqrt{9x^2-1}}+1}e^{3xt}\cosh\left(t\sqrt{9x^2-1}\right)
\end{equation*}
Furthermore
\begin{equation*}
D(x,t)=-\frac{t\sqrt{9x^2-1}}{e^{t\sqrt{9x^2-1}}}e^{3xt}
\end{equation*}
but
\begin{equation*}
\frac{1}{e^{t\sqrt{9x^2-1}}}=e^{t\sqrt{9x^2-1}}-2\sinh\left(t\sqrt{9x^2-1}\right)
\end{equation*}
then
\begin{equation*}
D(x,t)=t\sqrt{9x^2-1}\left(2\sinh\left(t\sqrt{9x^2-1}\right)-e^{t\sqrt{9x^2-1}}\right)e^{3xt}
\end{equation*}
and
\begin{equation*}
D(x,t)=t\sqrt{9x^2-1}\left(\sinh\left(t\sqrt{9x^2-1}\right)-\cosh\left(t\sqrt{9x^2-1}\right)\right)e^{3xt}.
\end{equation*}
Which means that
\begin{equation*}
D(x,t)=\left(9x^2-1\right)\sum_{n\geq0}nB^{*}_{n-1}(x)\frac{t^{n}}{n!}-\sqrt{9x^2-1}\sum_{n\geq0}
nC_{n-1}(x)\frac{t^{n}}{n!}.
\end{equation*}
Finally we obtain
\begin{equation*}
D_n(x)=n\left(9x^2-1\right)B^{*}_{n-1}(x)-n\sqrt{9x^2-1}C_{n-1}(x)
\end{equation*}
and the identity \eqref{eq2th2} follows.
\end{proof}
Frontczak \cite[Corollary 4]{Frontczak} proved the following
relations between balancing numbers and Bernoulli numbers
\begin{equation*}
\sum_{k=0\atop {n\equiv k \;{\rm(mod\;2)}}}^{n}{n\choose
k}32^{\frac{n-k}{2}}B^{*}_kB_{n-k}=nC_{n-1}
\end{equation*}
and
\begin{equation*}
\sum_{k=0\atop {n\equiv k \;{\rm(mod\; 2)}}}^{n}{n\choose
k}32^{\frac{n-k}{2}}\left(2^{n-k}-1\right)C_kB_{n-k}=8nB^{*}_{n-1}.
\end{equation*}
An improvement of these identities is given in the following
corollary.
\begin{corollary}
Let $n\geq1$. Then
\begin{equation}
\sum_{k=0}^{n}{n\choose
k}32^{\frac{n-k}{2}}B^{*}_kB_{n-k}=n\left(C_{n-1}-2\sqrt{2}B^{*}_{n-1}\right)
\end{equation}
and
\begin{equation}
\sum_{k=0}^{n}{n\choose
k}32^{\frac{n-k}{2}}\left(2^{n-k}-1\right)B_{n-k}C_k=8nB^{*}_{n-1}-2n\sqrt{2}C_{n-1}.
\end{equation}
\end{corollary}
\begin{proof}
Evaluate \eqref{eq1th1} and \eqref{eq2th2} at the point $x=1$ to get
the desired results.
\end{proof}



\begin{thebibliography}{10}

\bibitem{Apost} T. M. Apostol, 
{\it Introduction to Analytic Number Theory}, Springer, 1976.

\bibitem{Byrd} P. F. Byrd, Relations between Euler and Lucas numbers,
\emph{Fibonacci Quart.} \textbf{13} (1975), 111--114.

\bibitem{Frontczak} R. Frontczak, Relating Fibonacci numbers to Bernoulli
numbers via balancing polynomials,
\emph{J. Integer Sequences} \textbf{22} (2019)
\href{https://cs.uwaterloo.ca/journals/JIS/VOL22/Frontczak/front4.html}{Article 19.5.3}.

\bibitem{Frontczak2} R. Frontczak, On balancing polynomials,
\emph{Appl. Math. Sci.} \textbf{13} (2019), 57--66.

\bibitem{Frontczak3} R. Frontczak, Powers of balancing polynomials and
some consequences for Fibonacci sums, \emph{Inter. Jour. Math.  Anal.}
\textbf{13} (2019), 109--115.

\bibitem{VERN} V. E. Hoggatt, Jr., {\it Fibonacci and Lucas Numbers},
Fibonacci Association, 1971. 

\bibitem{Patel} B. K. Patel, N. Irmak, and P. K. Ray, Incomplete balancing and
Lucas-balancing numbers, \emph{Math. Rep.} \textbf{20} (2018),
59--72.

\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}: Primary
11B37; Secondary 11B65, 05A15.

\noindent \emph{Keywords:} Bernoulli number, balancing polynomial,
balancing number, Fibonacci number, Lucas number, Cauchy product of
generating functions.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A000032},
\seqnum{A000045},
\seqnum{A001109},
\seqnum{A100615}, and
\seqnum{A122045}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received December 12 2019;
revised versions received  February 27 2010; March 1 2020.
Published in {\it Journal of Integer Sequences}, March 18 2020.

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\noindent
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\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
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