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\begin{document} 

% This paper is published in %%
%{\fns Integers: \emph{Electronic Journal 
% of Combinatorial Number Theory} (2003)}





%\title{Jensen proof of a curious binomial identity}

%\author{\textbf{CHU Wenchang - DI CLAUDIO Leontina Veliana}}

%\subjclass{Primary 05A10, Secondary 05A19}



%\maketitle

\pagestyle{myheadings}
\markright{\smalltt INTEGERS: \smallrm 
ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 3 (2003),
\#A20\hfill}

\thispagestyle{empty}
\baselineskip=15pt
\vskip 30pt



%\vspace{-9mm}

%\centro{Dipartimento di Matematica, 
%Universit\a`{a} degli Studi di Lecce\\
%Lecce-Arnesano P. O. Box 193, 73100 Lecce, ITALIA\\
%chu.wenchang@unile.it -- leontina@email.it}



\begin{center}
{\bf JENSEN PROOF OF A CURIOUS BINOMIAL IDENTITY}
\vskip 20pt
{\bf Wenchang Chu }\\
{\smallit 
Dipartimento di Matematica, 
Universit\a`{a} degli Studi di Lecce,
Lecce-Arnesano, P. O. Box 193, 73100 Lecce, Italia}\\
{\tt chu.wenchang@unile.it}\\
\vskip 10pt
{\bf Leontina Veliana Di Claudio }\\
{\smallit Dipartimento di Matematica, 
Universit\a`{a} degli Studi di Lecce,
Lecce-Arnesano, P. O. Box 193, 73100 Lecce, Italia}\\
{\tt leontina@email.it}\\
\end{center}
\vskip 30pt
\centerline{\smallit Received: 5/22/03, Revised: 12/25/03, Accepted:
12/26/03, Published: 12/30/03 }
\vskip 40pt

\centerline{\bf Abstract}

\noindent{By means of the Jensen formulae 
on binomial convolutions, a new proof is presented 
for a curious identity due to Z.-W. Sun.}



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Based on double recurrence relations, 
Sun~\cite{kn:sun} discovered the following
binomial identity
\[S_m\::=\:
(x+m+1)\sum_{i=0}^{m}(-1)^i
\binom{x+y+i}{m-i}\binom{y+2i}{i}
-\sum_{i=0}^{m}(-4)^i
\binom{x+i}{m-i}
\:=\:(x-m)\binom{x}{m}.\]



Recently, three alternative proofs have been provided by Panholzer 
and Prodinger~\cite{kn:panho} via the generating function method, 
by Merlini and Sprugnoli~\cite{kn:merlini} through Riordan arrays,
and by Ekhad and Mohammed~\cite{kn:wz} based on the ``WZ'' method. 
Combining Jensen's identity and Chu-Vandermonde convolution formulae 
on binomial coefficients, we present yet another proof for this result 
which provides a shortcut.



By means of the Jensen formulae (cf.~\cite[Eq\:8]{kn:chu}
for example) on binomial convolutions
\[\sum_{i=0}^{m}\binom{a+bi}{i}\binom{c-bi}{m-i}
\:=\:\sum_{k=0}^{m}\binom{a+c-k}{m-k}b^k\]
the first binomial sum displayed in $S_m$ 
can be reformulated as 
\xalignz{
\sum_{i=0}^{m}(-1)^i\binom{x+y+i}{m-i}\binom{y+2i}{i}		
&=(-1)^m\sum_{i=0}^{m}\binom{y+2i}{i}
\binom{-1-x-y+m-2i}{m-i}\\
&=(-1)^m\sum_{k=0}^m\binom{-1-x+m-k}{m-k}2^k
\:=\:\sum_{k=0}^{m}\binom{x}{m-k}(-2)^k.}



For a complex $x$ and a natural number $n$, denote 
the shifted factorial of $x$ of order $n$ by 
\[(x)_0=1\quad\text{and}\quad 
(x)_n=x(x+1)\cdots(x+n-1)
\quad\text{for}\quad n=1,2,\cdots.\] 
In accordance with the parity of $k$, writing 
$k:=\delta+2k'$ with $\delta:=0,1$ and then 
performing the replacement $j:=i-\delta-k'$, 
we can derive the following binomial 
coefficient identity:
\bnm
\sum_{\frac{k}{2}\le i \le k}
\binom{i}{k-i}(-4)^i
&=&(-4)^{\delta+k'}\sum_{j=0}^{k'}
\binom{\delta+k'+j}{k'-j}(-4)^j\\
%\:=\:
&=&(-4)^{\delta+k'}\sum_{j=0}^{k'}
\frac{(-1)^j(\delta+k'+j)!}
     {j!(k'-j)!(1/2+\delta)_j}\\
&=&(-4)^{\delta+k'}
\frac{(1+\delta)_{k'}}{(1/2+\delta)_{k'}}
\sum_{j=0}^{k'}\binom{-1-\delta-k'}{j}
\binom{\delta-1/2+k'}{k'-j}\\
&=&(-4)^{\delta+k'}
\frac{(1+\delta)_{k'}}
     {(1/2+\delta)_{k'}}
\binom{-3/2}{k'}
\:=\:(-1)^k2^k(1+k)
\enm
where the Chu-Vandermonde convolution 
formulae has been applied.



This binomial identity allows us to express 
the second sum displayed in $S_m$ as
\bnm
\sum_{i=0}^{m}\binom{x+i}{m-i}(-4)^i
&=&\sum_{i=0}^{m}(-4)^i
\sum_{k=i}^{m}\binom{x}{m-k}\binom{i}{k-i}\\
&=&\sum_{k=0}^{m}\binom{x}{m-k}
\sum_{\frac{k}{2}\le i \le k}
\binom{i}{k-i}(-4)^i\\
&=&\sum_{k=0}^{m}\binom{x}{m-k}(-2)^k(k+1).
\enm



Therefore the linear combination of the 
two binomial sums in $S_m$ results in
\xalignz{
S_m&=\sum_{k=0}^{m}(x+m-k)\binom{x}{m-k}(-2)^k
\:=\:\sum_{k=0}^{m}\big\{(x-m+k)+2(m-k)\big\}
\binom{x}{m-k}(-2)^k\\
&=\sum_{k=0}^{m}(1+m-k)\binom{x}{1+m-k}(-2)^k
-\sum_{k=0}^{m}(m-k)\binom{x}{m-k}(-2)^{k+1}\\
%\:=\:
&=(m+1)\binom{x}{m+1}}
where the two sums with respect to $k$ in the last line 
have been telescoped. This completes the proof of the 
identity originally due to Sun. 





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\begin{thebibliography}{99}
\newcommand{\btm}[4]{\bibitem{kn:#1}
           {#2,}~{\em #3,}~{#4.}}		



\btm{chu}{W. Chu}
    {On an extension of a partition identity 
     and its Abel analog}
    {J. Math. Research \& Exposition
     6:4 (1986), 37-39; MR89b:05021}



\btm{wz}{S. B. Ekhad and M. Mohammed}
    {A WZ proof of a ``curious'' identity}
    {Integers: The Eletronic Journal of Combinatorial 
     Number Theory 3 (2003), A6: 2pp}



\btm{knuth}{R. L. Graham, D. E. Knuth and O. Patashnik}
    {Concrete Mathematics}
    {Addison-Wesley Publ. Company, 
     Reading, Massachusetts, 1989}



\btm{merlini}{D. Merlini and R. Sprugnoli}
    {A Riordan array proof of a curious identity}
    {Integers: The Eletronic Journal of Combinatorial 
     Number Theory 2 (2002), A8: 3pp}



\btm{panho}{A. Panholzer and H. Prodinger}
    {A generating functions proof of a curious identity}
    {Integers: The Eletronic Journal of Combinatorial 
     Number Theory 2 (2002), A6: 3pp}



\btm{sun}{Z.-W. Sun} 
    {A curious identity involving binomial coefficients}
    {Integers: The Eletronic Journal of Combinatorial 
     Number Theory 2 (2002), A4: 8pp}



\end{thebibliography}



\end{document}