%%  This article has been submitted and accepted for publication in
%%  THE FOATA FESTSCHRIFT, 
%%  a special issue of THE ELECTRONIC JOURNAL OF COMBINATORICS
%%  dedicated to DOMINIQUE FOATA at the occasion of his 60th birthday.
%%
%%  Author(s):		SHALOSH B. EKHAD AND J.E. MAJEWICZ
%%  Title:		A SHORT WZ-STYLE PROOF OF ABEL'S IDENTITY
%%  Date of submission:	Februaray 17, 1995
%%  TeX version:	AMSTeX
%%  File size (approx):	3 kB
%%  Number of pages:	1
%%  Author's email:	ekhad@euclid.math.temple.edu
%%
%%
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\topmatter
\title  A short WZ-style proof of Abel's identity \endtitle
\author  Shalosh B. Ekhad and John E. Majewicz \endauthor
\address  Department of Mathematics, Temple University, Philadelphia, PA
        19122\endaddress
\date Submitted: February 17, 1995; Accepted: July 25, 1995\enddate
\abstract
Using a certification procedure for Abel-type sums, we present a computerized
         proof of Abel's identity.  \endabstract
\email  ekhad\@math.temple.edu, jmaj\@euclid.math.temple.edu \endemail
\dedicatory  Dedicated to Dominique Foata on the occasion of his
         sixtieth birthday.  \enddedicatory
\endtopmatter
 
\document
Abel's identity \cite{1},\cite{2} can be proved in many ways, 
including the elegant combinatorial methods of \cite{3}.  
Here we present the first computer-generated proof using 
the methods introduced in \cite{4}.
 
\proclaim{Theorem(\cite{1}; \cite{2}, p.128)} For $n\ge0$:
$$
\sum_{k=0}^n\binom{n}{k}(r+k)^{k-1}(s-k)^{n-k}=\frac{(r+s)^n}{r} \, .\tag1
$$
\endproclaim
 
\demo{Proof}  Let $F_{n,k}(r,s)$ and $a_n(r,s)$ denote, respectively, 
the summand and sum on the LHS of (1), and let 
$G_{n,k}:=(s-n)\binom{n-1}{k-1}(k+r)^{k-1}(s-k)^{n-k-1}$. Since
$$
F_{n,k}(r,s)-sF_{n-1,k}(r,s)-(n+r)F_{n-1,k}(r+1,s-1)
        +(n-1)(r+s)F_{n-2,k}(r+1,s-1)=G_{n,k}-G_{n,k+1} \,,
$$
(check!), we have by summing from $k=0$ to $k=n$, thanks to the telescoping
on the right:
$$
 a_n(r,s)-sa_{n-1}(r,s)-(n+r) a_{n-1}(r+1,s-1)+
        (n-1)(r+s) a_{n-2}(r+1,s-1)=0.
$$
Since $(r+s)^n\cdot r^{-1}$ also satisfies 
this recurrence (check!) with the same initial 
conditions $a_0(r,s)=r^{-1}$ and 
$a_1(r,s)=(r+s)\cdot r^{-1}$, $(1)$ follows.\qed  
 
We thank Herb Wilf
for a great shrinking comment on an earlier (much longer) version.
\enddemo
 
 
\Refs 
 
\ref \no 1 \by N. Abel \pages 159-60  
\paper  Beweis eines Ausdruckes, von welchem die Binomial-Formel ein einzelner Fall ist  \yr 1826 \jour  Crelle's J. Mathematik  \endref
 
\ref \no 2 \by Louis Comtet  
\book Advanced Combinatorics \publ  D. Reidel Publ. Co.
\publaddr Dordrecht/Boston \yr 1974 \page 128 \endref
 
\ref \no 3  \by D. Foata \pages 181-186
\paper Enumerating k-Trees \yr 1971 \jour Disc. Math. \endref
 
\ref \no 4  \by J. Majewicz
\paper  WZ-type certification procedures and Sister Celine's technique for
         Abel-type sums, 
\jour to appear (also available via anonymous ftp to
         ftp.math.temple.edu in directory pub/jmaj ) \endref
 
\endRefs
 
 
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