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%RIGOROUS EXPERIMENTAL COMBINATORICS
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%\def\LeftHead{ \centerline{Doron Zeilberger}}
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\centerline{\bf 
Report on Sabbatical Leave Fall 2007
}
\bigskip  
\centerline{ \it Doron Zeilberger}
\medskip 
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019.
{\tt zeilberg@math.rutgers.edu} \hfill\break  
{\tt http://www.math.rutgers.edu/\Tilde zeilberg/} .
\bigskip 

{\bf Summary of Results 0401124}
 
I continued to practice
a new research methodology, that can be loosely
called {\it rigorous experimental mathematics}.
It has something in common with both ``mainstream''
experimental mathematics, as preached by the Borwein
brothers, David Bailey, Victor Moll, and their collaborators,
and  {\it automated theorem proving} 
but is definitely distinct from them.
It is based on what I call {\it the ansatz ansatz}.

One ``teaches'' the computer how to ``conjecture an answer''
to a problem, and then ``teaches''
that very same computer to prove its own conjectures rigorously.
The novelty is that {\it both} the 
conjecturing and the proving are automatically done
by the computer. This does not mean that human mathematicians
are superfluous. Quite the contrary!
Someone has to ``teach'' the computer, i.e.
design algorithms and meta-algorithms for both proving and
conjecturing. In my experience,
this act of ``teaching'' the computer how
to do mathematics is at least as challenging as doing
mathematics ``by hand'', and in my humble opinion, time
much better spent, since the vast potential of the computer
is still very underutilized and underrated,  and it is
important to have mathematicians, like myself and my 
students, who are dedicated to that activity, that
I believe will soon revolutionize mathematics.

In particular, The so-called WZ theory was further extended and
applied, in collaboration with my Chinese colleagues,
whom I visited during my sabbatical leave 2007.



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