
%opa.tex: Liebe Opa Paul, Ich Bin Auch Ein Experimental Scientist!
%
%a Plain TeX file by Doron Zeilberger (11 pages)
 
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
 0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
 height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6 
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\parindent=0pt
\overfullrule=0in
%end macros
 
\bf
\centerline
{
Liebe Opa Paul, Ich Bin Auch Ein Experimental Scientist!
}
\rm
\bigskip
\centerline{ {\it 
Doron ZEILBERGER
}\footnote{$^1$}
{\eightrm  \raggedright
Department of Mathematics, Rutgers University (New Brunswick),
Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway,
NJ 08854-8019, USA.
%\break
{\eighttt zeilberg@math.rutgers.edu} \hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/   .}
First version: Oct.  9, 2002. 
Edited version of the transcript of a talk 
(entitled: `Determinants and Computer Algebra')
given at the Max Planck Institute,
Leipzig, Aug. 5, 2002, 15:00-16:00.
Accompanied by the Maple package {\tt CLD} available from
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/programs.html}
Supported in part by the NSF. 
} 
}

\qquad\qquad\qquad\qquad\qquad
{\it Dedicated to the memory of my grandfather Paul Alexander}
 
{\bf Abstract:}  Contrary to popular belief, Math {\it is} 
an experimental science.
Hence I am an experimental scientist, just like my beloved
grandfather, Dr. Paul Alexander 
(1870-1942, Dr. Phil. Rer. Nat., Chemie,  1897).

{\bf Leipzig}

This is my first visit to Leipzig. My main reasons for
coming here are {\it personal}: to look up the graves and dwellings of
my great-grandparents, Salomon and Rebecka Alexander, and to explore 
the city and the university where my grandfather, Paul Alexander, 
grew up and studied. But I thought that it would be nice to combine the
business of family pilgrimage with the pleasure of giving a math talk.
I am very grateful to Wolfgang Hackbusch 
for kindly accepting my offer to give a seminar talk here. 
I am also very thankful to all of you for coming!

Even though this talk is supposed to be on {\it math}
rather than on {\it family history},
let me nevertheless spend a few minutes telling you about my
grandfather Paul Alexander, to whom this talk is dedicated.

{\bf A Few Words about Paul Alexander}

1. Paul got his Dr. Phil. in 1897, here in Leipzig. His
dissertation was entitled:
``{\it \"Uber die Einwirkung von o-Nitrobenzylchlorid auf
Natriummalons\"aure\"athylester}''.

2. Paul was the inventor of efficient {\it verfahren}s for
the regeneration (recycling) of caoutchouc (rubber).

3. Paul had many patents, e.g. US \# 844077, issued in 1907
entitled ``{\it Process for the production of aqueous caoutchouc
solution and the regeneration of rubber waste}''.

4. He contributed several articles to the famous Ullmann
{\it Enzyklop\"adie} of Industrial Chemistry.

Paul's academic {\it vater} was
the great chemist {\bf Johaness Wislicensus} (1835-1902),
who  made many important contributions to chemistry. One was
the suggestion that there are {\it geometrical isomers} 
exemplified by the two forms of the lactic acid. He inspired,
and later enthusiastically endorsed the revolutionary theory of
Le-Bell and J. Van t' Hoff (who got the very first chemistry
Nobel, in 1901). He was, most probably, a very nice guy!

But not {\it all} Leipzig professors were so nice. One
especially nasty specimen was the great organic chemist {\bf Hermann Kolbe},
who was Wislicensus's predecessor at the Leipzig chemistry chair.
Kolbe, known for his very acerbic wit, 
commented on the Le-Bell-van t' Hoff theory as follows:

{\it `` .... There is an overgrowth of the weed of the
seemingly learned and ingenious but in reality trivial and
stupefying natural philosophy... which had been dressed up
in modern fashion and rouged freshly like a whore whom one tries
to smuggle into good society where she does not belong...''}

It would be unfair to ridicule Kolbe for the
{\it substance} of his critique, since now we have the benefit of hindsight,
and it is not his fault that history proved him wrong.
Science does need its share of conservatives 
to serve as bouncers to guard us against wild speculations
like cold fusion and organic transistors.
Nevertheless, one can be critical without being mean, and often
we are critical because we feel like being mean.

Controversy is much more prevalent in science than it is in math,
but even math has its share. As late as 1903 there were still people
who did not accept non-Euclidean Geometry. We all heard about
the Kronecker-Cantor and Hilbert-Brouwer feuds.
More recently we witnessed the heated debate
concerning  the role of `theoretical math', 
as well as the Andrews-Zeilberger mini-controversy about
semi-rigorous mathematics. As math will become more scientific,
we should expect more
controversy, which will make things more interesting!

Yet another Leipziger was {\bf Hermann Hankel}, who said:

``{\it
In most sciences one generation tears down what another has
built, and what one has established, another undoes. In
MATHEMATICS ALONE each generation adds a new storey to the old
structure}.''

In other words, math is a peaceful, non-violent, cumulative
enterprise. Hankel also expressed another ``obvious truth'' about our
profession:

``{\it Mathematics is purely intellectual, a pure theory of forms,
which has for its objects not the combination of quantities
and images, but things of thought to which there 
{\bf could} correspond
effective objects or relations, even though such a correspondence
is {\bf not necessary} }.''(my emphasis)

But even though Hankel was probably much nicer than Kolbe,
his statements too are starting to be wrong. Because of 
the {\bf Computer},
Mathematics is becoming an empirical, descriptive, and experimental science,
just like Chemistry! `Modern' Math, that
is supposedly {\it a priori}, will soon join the ranks of
Aristotelian physics that was also {\it a priori}.
Unlike the latter, however, it will not be labelled false,
but, perhaps worse, would be considered utterly trivial,
since computer-generated math will be able to discover, and
prove, much deeper results.

The first mathematical area that took advantage of the computer revolution was
{\bf Numerical Math}. Here is a quotation form yet another 
Leipziger, do you know who?, that in a very major International Congress said:

``{\it In the former times there were obvious reasons why
$n$  was rather small. This
is why {\bf Numerical Math} did not appear as a discipline of
its {\bf own} before the help of electronic {\bf computers}
was available}.''

That same person also said, later at the same talk:

{\it ``Large scale computations are those which are 
{\bf almost} too large to be computed on {\bf present}
machines.''} (emphasis added).

I am sure that most of you will recognize these words as
belonging to your esteemed colleague, Wolfgang
Hackbusch, who in his insightful plenary ICM 1998 talk
presented the state of the art in numerical math and
scientific computation, and stated that in order
to be able to solve very large problems, one has to make
some {\it compromises}. The first compromise is to
abandon the {\it exact} and settle for the {\it approximate}
(what he called $\epsilon$-oriented). Another compromise
is to abandon the {\it general} and settle for the
{\it special}, i.e. trying to solve special classes that
often come up in practice, e,g. {\it sparse} systems,
or his own favorite, $H-$ systems.

He also talked about {\bf Algorithmic Paradigms}.
In particular, about
{\bf Hierarchy}, {\bf Adaptivity}, and {\bf (De)composition}.

All this sounds like a {\it sound} methodology for a {\it science}
that has a strong empirical and experimental flavor. In addition,
numerical mathematicians do {\it numerical experiments} on a
regular basis to test their algorithms, and the empirically
observed  performance is often much better than the theoretical,
{\it a priori}, prediction.

I strongly feel that Hackbusch's talk [H] should be required reading to
all {\it pure} mathematicians, especially to those, like myself,
who try to get as much as possible out of the computer. We too
should be able to develop {\it algorithmic paradigms}
and {\it research methodologies} of our own. A good start
is by adapting to our needs the already acquired wisdom from numerics,
as described by Hackbusch.

So far the use of computers in `pure' math, with a few exceptions,
was rather {\it methodologically}
boring. It consisted mainly in testing conjectures. 

Much more interesting, from the {\it methodological} point of view, are
{\it computer-assisted proofs}. The most famous being the Appel-Haken
Four Color Theorem. Here there was a {\it human-machine dialog} that
helped {\it design} the proof, and once the proof was proposed
(a certain explicit unavoidable set of reducible configurations),
it was verified by computer.

There is also a rapidly growing effort in {\it automated proving}.
These can be roughly divided to {\it logic-based} programming,
pursued in AI and computational logic, that uses {\it resolution},
{\it tableaux} and other methods, and {\it ansatz-based} programming,
in which the objects are known to belong to a well-defined
algebraic class that possesses {\it canonical} forms, or at least
{\it normal forms}, making it decidable whether $A=B$ or
$A=0$ respectively. 

A  famous example is {\it Euclidean Geometry}, in which,
thanks to Rene Descartes, the objects are {\it rational functions}
in the parameters, and in more complicated situations
{\it ideals}, for which the Buchberger algorithm supplies
a canonical form.
There is also
{\bf WZ theory}, that is an {\it algorithmic proof theory}
for {\bf hypergeometric summation and integration} identities.
However, in both these cases the algorithms themselves were 
created by humans, and while it is true that they can prove
many results that previously required ad hoc human proofs,
the very existence of these algorithms makes these `computer-generated
proofs', and hence also the results that they prove, trivial
in some sense, since we are {\it guaranteed} to get a proof or
refutation, time- and space- limitations permitting.

But what makes research so exciting is that it is
a {\bf gamble}. You don't know, beforehand, whether you will
succeed or fail. You also want to allow for
{\it serendipity}, the possibility that
in your computer's attempts to prove Goldbach 
it will discover something even
more interesting. So what we desperately need are
{\bf Algorithmic Paradigms for Computer-Generated Research.}

In other words, we need methodologies for creating new algorithms
that will enable computers to discover, and {\it prove}, new results,
without knowing, beforehand, whether it will succeed, but with
a fair chance that it will.

For the sake of simplicity, let's focus on proving
{\it identities}.
These are mathematical statements whose format is $A=B$.
The traditional way is to try and manipulate $A$,
finding another object $A_1$ that `looks different'
but is really the same. The problem is that
there are usually several choices. Then one can
try to find $A_2$, getting a string
$A=A_0=A_1=A_2= \dots $, and if in {\it luck}, or one
has a good intuition, or the problem is not very deep,
one gets to $B$. Since at every stage there are several
choices, and there is no upper bound for the number of steps,
this method leads to exponential explosion. 

If both $A$ and $B$ belong to a class of mathematical
objects for which there exists a {\it canonical form},
and there is also an {\it algorithm} $A \rightarrow c(A)$,
for reducing any object to its canonical form, then all one has to do is
compute $c(A)$ and $c(B)$ and see whether they are the same.

But what if you don't know $B$? In other words, you have
an input $A$ that is `ugly', and belongs to a 
{\bf general ansatz}, but you hope that there exists
a {\bf nice} $B$, such that $A=B$. By `nice' I mean
belonging to a suitable {\bf subansatz}, a {\bf specific ansatz}.
Can the computer find $B$? (if it exists?).

Let's assume that the objects of the {\bf specific ansatz}
have a natural complexity, such that one can
express the objects of any given, finite, complexity, in
generic form, with {\it undetermined} coefficients.

To make this non-trivial, we must assume that $A$ and $B$
are really infinite classes, i.e. $A(n)$ and $B(n)$, where
$n$ is an integer parameter. We assume that for any
specific $n_0$, $A(n_0)$ and $B(n_0)$ are computable,
where the latter is in terms of undetermined coefficients.
Then by plugging-in enough values for $n_0$ and
solving the system $A(n_0)=B(n_0), n_0=1,2, \dots L$,
for sufficiently large $L$, for the unknown undetermined
coefficients of $B(n)$, and if the computer
finds a solution, then we have a genuine new
theorem, that the computer discovered from scratch.
Once conjectured, it should be routine to
prove that $A(n)=B(n)$ by plugging into the
defining equation of $A$ in the {\bf general ansatz}.

Many times it is not possible to prove that $A(n)=B(n)$
directly. Then one looks for more general objects
$A'(n,r)$, that does belong to a general ansatz, and
such that $A(n)=A'(n,0)$. 
If that general ansatz contains a subansatz of nice objects
$B'(n,r)$, one may try to find it, prove algorithmically
that $A'(n,r)=B'(n,r)$, and finally deduce that
$A(n,0)=B'(n,0)$, where $B'(n,0)$ is nice since $B'(n,r)$ is.

If the above seems a bit vague, I hope that the case-study below,
of {\bf automated} (symbolic) determinant-evaluation, using
the {\bf Dodgson ansatz} for the {\it general} ansatz, and
the {\it hyperhypergeometric} ansatz for the {\it specific}
(nice) ansatz , would make this approach crystal clear.

{\bf Trying to Abstract from the Well-Known
Explicit Evaluation of the Determinant of the Hilbert Martix}

The Hilbert matrix
$$
A(n)_{i,j}:={{1} \over {i+j+1}} \quad , \quad 0 \leq i,j \leq n-1 \quad ,
$$
is dear to numerical analysts because it is a famous example
of a badly-conditioned matrix.
Its determinant has a well-known explicit evaluation
$$
\det (A(n))= \prod_{i=1}^{n-1} {{i!^4} \over {(2i+1)!(2i)!}} \quad .
$$
Let's call the right-hand side $b(n)$. What's nice about
$b(n)$ is that the ratio: $c(n):=b(n+1)/b(n)$, 
is a hypergeometric sequence namely,
$c(n)=n!^4/((2n+1)!(2n)!)$. But a {\it hypergeometric sequence}
is precisely one whose consecutive ratio is a rational function.
In this case $d(n):=c(n)/c(n-1)$ equals 
$n^2/(4(2n+1)(2n-1))$.

How would {\it anyone} start to prove it? A natural way would be
by induction on $n$.  However, 
having only one parameter is too restrictive.

My favorite way to evaluate determinants ([Z1][Z2]) is

{\bf Reverend Charles Lutwidge Dodgson's Determinant
Condensation Rule}

It states the following.
For any $n$ by $n$ matrix $A$, let $A_r(i,j)$ denote the $r$ by $r$ 
minor consisting 
of $r$ contiguous rows and 
columns of $A$, starting with row $i$ and column $j$. In particular, 
$A_n(1,1) = \det A$. Then, according to Dodgson [D],
$$
 A_n(i,j) A_{n-2}(i+1,j+1)  =
 A_{n-1}(i,j) A_{n-1}(i+1,j+1) -  A_{n-1}(i+1,j) A_{n-1}(i,j+1).
\eqno(Lewis)
$$
The desired determinant is $A_n(1,1)$.
In many cases, $A_n(i,j)$ turns out,
conjecturally at first, to have an explicit expression, 
involving single and double products. Whenever this is the case
the proof of the conjectured evaluation is completely
routine, 
by induction on $n$, by checking that $(Lewis)$ is satisfied
by that conjectured expression,
and by checking the trivial initial conditions for $n=0$ and
$n=1$. Finally, to get an explicit expression for the
original determinant, all one has to do is plug in $i=1$ and
$j=1$.

In [AE2] this method was used to get computer-assisted proofs
of numerous determinant identities. But my goal is to make
things {\it completely} automatic, and human-free.

To keep things simple (after all, the main point here is to
introduce a {\it research methodology}, not to find exciting
new results), let's focus on Hankel matrices, which 
have the form $(h(i+j))$ for some
sequence $h(r)$ (that for us would have to be an explicit expression).

So given a discrete function $h(r)$ (say a hypergeometric sequence),
we have the  {\bf General Problem} 
of evaluating the $n \times n$ determinant
$$
A(n,r):=\det (h(r+i+j)), 0 \leq i,j \leq n-1 \quad .
\eqno(Hankel)
$$

Even if we are only interested in $A(n,0)$, we still need 
the $r$, as will become apparent shortly.
For Hankel matrices $(Hankel)$, Dodgson's rule specializes to:
$$
A(n,r)={{A(n-1,r)A(n-1,r+2)-A(n-1,r+1)^2} \over
{A(n-2,r+2)}} \quad .
\eqno(HankelDod)
$$
Now, in some sense, this is {\it already} an answer, since it displays
$A(n,r)$ in the ansatz of double sequences satisfying
{\it partial non-linear recurrence equations with constant coefficients}.
Indeed since $A(0,r)=1$ and $A(1,r)=h(r)$, 
$(HankelDod)$, gives a quick way to crank out the sequence
$A(n_0,r)$ for $n_0=0,1,2, \dots, N_0$ for any desired
$N_0$. Of course, one can argue that the very definition is
already an `answer' just declare the  class of determinants of
hypergeometric determinants a legitimate ansatz! But
we would like to do better.

Inspired by the Hilbert matrix, for which $A(n,0)$ turned out
to be hyperhypergeometric in $n$, it turns out (experimentally,
at first), that $A(n,r)$ also has this property for each $r$.
Not only that, the ratio-of-ratios 
$(A(n,r)/A(n-1,r))/(A(n-1,r)/A(n-2,r))$ is not only a rational
function of $n$, but of {\it both} $n$ and $r$.
Furthermore, it also turns out that it is also
hyperhypergeometric in $r$, i.e.
$(A(n,r)/A(n,r-1))/(A(n,r-1)/A(n,r-2))$  is another rational function
of $(n,r)$. Finally the `mixed-ratio'
$(A(n,r)/A(n-1,r))/(A(n,r-1)/A(n-1,r-1))$ is also a rational
function of $(n,r)$.

{\bf A New Ansatz is Born: Hyperhypergeometric Double-Sequences}

{\bf Definition}: A double sequence $B(n,r)$ is
{\it hyperhypergeometric}  if the three discrete functions
$$
B_{11}(n,r):={{B(n,r)B(n-2,r)} \over {B(n-1,r)^2}} \,,\,
B_{12}(n,r):={{B(n,r)B(n-1,r-1)} \over {B(n-1,r)B(n,r-1)}} \,,\,
B_{22}(n,r):={{B(n,r)B(n,r-2)} \over {B(n,r-1)^2}} \, ,
$$
are all rational functions of $(n,r)$. Hence, hyperhypergeometric
double-sequences may be identified with 

{\bf Triples of Rational Functions} $(B_{11},B_{12},B_{22})$ satisfying
the obvious { \it compatibility conditions}:
$$
{{B_{11}(n,r)} \over {B_{11}(n,r-1)}} =
{{B_{12}(n,r)} \over {B_{12}(n-1,r)}} \quad , \quad
{{B_{22}(n,r)} \over {B_{22}(n-1,r)}} =
{{B_{12}(n,r)} \over {B_{12}(n,r-1)}} \quad .
\eqno(Compatibilty)
$$
In addition we have to specify the initial conditions
$b_{00}=B(0,0), b_{01}=B(0,1), b_{10}=B(1,0)$.

So suppose you have a conjectured hyperhypergeometric expression $B(n,r)$ for
the family of Hankel determinants 
$A(n,r):=\det (h(r+i+j)), 0 \leq i,j \leq n-1$.
By Dodgson's rule, it is enough to verify that
$B(0,r)=1$ $B(1,r)=h(r)$, and
$$
B(n,r)={{B(n-1,r)B(n-1,r+2)-B(n-1,r+1)^2} \over
{B(n-2,r+2)}} \quad .
\eqno(HankelDod')
$$
By taking ratios, this is equivalent to, in terms
of the rational functions $B_{11},B_{12},B_{22}$:
$$
{{B_{12}(n-1,r+2)B_{12}(n-1,r+1)} \over {B_{11}(n,r)}}-
{{B_{12}(n,r+2)B_{12}(n-1,r+2)} \over {B_{22}(n,r+2) B_{11}(n,r+1)}}=
1 \quad ,
\eqno(VerifyHankelDod)
$$
which Maple (or Mathematica, etc.) can verify routinely, and
hence {\it prove} the conjecture.

But what about {\it discovering} the identity in the first place?
{\bf Can a computer do that? All by itself?}

{\bf You Bet It Can!}. Now that we have a well-defined {\it haystack}, the
ansatz of hyperhypergeometric double-sequences, we can let
the computer compile a table of $A(n,r)$ for $n,r \leq L$ for
some finite $L$, either by using the determinant definition,
or more efficiently, by using $(HankelDod)$, starting with
$A(0,r)=1, A(1,r)=h(r)$. Then we let our beloved computer
compute the iterated ratios
$$
A_{11}(n,r):={{A(n,r)A(n-2,r)} \over {A(n-1,r)^2}} \,,\,
A_{12}(n,r):={{A(n,r)A(n-1,r-1)} \over {A(n-1,r)A(n,r-1)}} \,,\,
A_{22}(n,r):={{A(n,r)A(n,r-2)} \over {A(n,r-1)^2}} \, .
$$
for $2 \leq n,r \leq L$.

Then, assuming that $A(n,r)$ is indeed hyperhypergeometric, we
put $B_{11},B_{12},B_{22}$ in generic form for rational
functions in $(n,r)$ with {\it undetermined coefficients}, 
where the top and bottom of each are generic polynomials of
a guessed degree $d$. Now by plugging-in, we have $(L-1)^2$ equations
$$
B_{11}(n_0,r_0)-A_{11}(n_0,r_0)=0 \quad, \quad 2\leq n_0,r_0\leq L \quad .
$$
Clearing denominators, and setting the numerator equal to
zero, will give us a system of {\it linear} equations in the
unknown `undetermined' coefficients of $B_{11}$.  Similarly for
$B_{12}$ and $B_{22}$.
If the computer 
finds a solution, then we are done! If it does not, we can make
the guessed degree one higher, and try again. We can keep upping
the degree until we succeed or give up. Of course, no one said
that $A(n,r)$ must be hyperhypergeometric, it was only our
{\it conjecture} that it might. So humans still have
to {\it decide} which ansatzes to try, but once that decision is
made, and the program already exists, the computer does {\it everything}
from  $\alpha$ to $\omega$: conjecture the expression
$B(n,r)$ (in its equivalent form as the triple of rational functions
$B_{11}(n,r),B_{12}(n,r),B_{22}(n,r)$), and then proves it,
all by itself! Finally, it also verifies the compatibility
conditions $(Compatibility)$, which  also consist of routine
manipulations of rational functions.

{\bf Toeplitz Determinants}

As far as I know, Otto Toeplitz was not a Leipziger, and hence
it is unlikely that he ever bumped into my grandfather Paul
in the street or cafeteria. But he is still dear to me, 
in part because his widow was my nanny between the ages
of $0$ and $1$, and in part because I like his determinants,
that have the form $det(h(i-j))$. All we said above
about Hankel determinants carries over, with obvious modifications,
to Toeplitz determinants. For details see the source-code in the
Maple package {\bf CLD}, described below.

{\bf A User's Manual for the Maple package CLD}

(CLD stands for Charles Lutwidge Dodgson). First download
it from my website, by going to my homepage
(search Google for ``Zeilberger'' (or even for ``Doron'') or
\hfill\break
type {\tt http://www.math.rutgers.edu/\Tilde zeilberg/}) then
click on {\tt programs}, then click on {\tt CLD}.
Alternatively, just download 
{\tt http://www.math.rutgers.edu/\Tilde zeilberg/tokhniot/CLD}.

Once my Maple package {\tt CLD} is in your own computer,
stay in the same directory,
go into Maple, by typing {\tt maple}, or {\tt xmaple}, or
by clicking on the Maple icon. Once in Maple, type:
{\tt read CLD} (if you decide to go to a different directory,
you need the full path name of the file {\tt CLD}).

Now, all you have to do is follow the {\it on-line} instructions.
In particular, typing {\tt ezra();} will give you a list of
all the main procedures, i.e. those that you are likely to use.
Typing {\tt ezra1();} will give a list of {\it all } procedures,
so that {\bf you} can understand what is going on, and will be
able to improve and extend this rudimentary program to more general
classes of determinants and to explore other, more general, or completely
different, ansatzes.

The main procedures are {\tt EvalH} for the automatic discovery and
proof of Hankel-determinant-evaluations, and the Toeplitz analog
{\tt EvalT}. These give you the output in terse style. If you want
a math paper, ready for submission,  use the {\it verbose}
versions, {\tt EvalHpaper} and {\tt EvalTpaper}.

For example, typing the $28$-character string
{\tt EvalHpaper(r!*(m-r)!,n,r,4):}
would, after a couple of minutes (on my rather slow computer), output
a paper that does {\tt ALL} the following steps, previously
done by humans, with only some machine help, that lead to [AE1].

{\bf 1}. Conjecture the expression (this was first done
by the smart humans Greg Kuperberg and Jim Propp).

{\bf 2}. Prove it, by human-machine interaction (previously done by
Human Tewodros Amdeberhan and Machine Shalosh B. Ekhad).

{\bf 3}. Write up the paper [AE1] for publication (formerly done
by the human-partner of the A-E collaboration).

For the celebrated MacMahon determinant[M]
(very important in plane-partition enumeration, first proved by
the great Percy MacMahon, and then reproved by me[Z1], using
Dodgson's rule (with help from Ekhad)), type:
{\tt EvalTpaper(1/(m+r)!,n,r,4):}.

Finally, for a completely automated 
performance of all the phases of mathematical activity:
conjecture-proof-writing\_it\_up,
for the closed-form evaluation of the Hilbert matrix,
type: \hfill\break
{\tt EvalHpaper(1/(r+1),n,r,4):}.

{\bf Sample Input and Output Files}

The webpage of this paper (clickable from my homepage)
contains sample input and output files.

{\bf How to be Immortal}

Dying is a stupid reason to stop publishing. If you are lucky,
someone might find your unfinished work, finish it up, and
publish it as joint work (like Bruce Berndt did to B.M. Wilson).
But this will, at best, get you at most one or two posthumous papers.
What if you want to keep on publishing papers for ever?
Easy! First make your system-administrator promise not to close
your account after your demise. Then in the Maple package,
have a `Unix-escape' shell-program that submits the paper
to one of the many electronic journals. It might be a good idea
to sign an inclusive copy-right-transfer form for all your future
submissions.

Now write an {\it infinite} do-loop, with increasingly more
complicated determinants to be evaluated. Most of them will turn
out not to fit the given ansatz (in this case the hyperhypergeometric
ansatz), but whenever it does, and the computer succeeded in conjecturing,
and then, automatically proving it, the computer can
completely automatically, also do the {\bf submission}.

You can do even more! Suppose that after $100$ papers on this
subject, the editor finally decides to reject your 
$101^{th}$ posthumous paper, because it is `not interesting'. 
Then you can automatically send an angry
rebuttal. The variations are endless.

{\bf Suggestions for Further Work}

It should be relatively painless to do the $q$-analog of this,
and to also deal with determinants of general matrices
$(a(i,j))$ ($1 \leq i ,j \leq n$), i.e. not necessarily
Hankel or Toeplitz. Now we would have a 3-parameter
discrete function $A(n,r,s):=det(A(i+r,j+s)), 0 \leq i,j \leq n-1$,
and the appropriate ansatz
would be hyperhypergeometric sequences for {\it triple}
sequences.

Recall that $B(n,r)$ is {\it hyperhypergeometric} means that
$B(n,r)/B(n,r-1)$ and $B(n,r)/B(n-1,r)$ are hypergeometric.
This naturally leads to the more general ansatz for which
the above two ratios are $P$-recursive. Even more generally
we can consider solutions of linear recurrence 
whose coefficients are $P$-recursive (holonomic). I am sure
that with these more general ansatzes, many more determinants
will be computer-evaluable.

{\bf Why is This Exciting?: The Medium is The Message!}

With all due respect to the {\it substance} of this research,
i.e. determinant-evaluation, what makes this endeavor 
{\it so} exciting is the {\it form} and the
{\it research methodology}, of doing
{\it purely theoretical} and {\it completely rigorous} mathematics
using {\it experimental methods}. Of course, these are just
crude and clumsy beginnings, but as we, and the computer, will get
more experienced, this methodology will be applicable to proving
{\it Goldbach}, {\it RH}, {\it Navier-Stokes}, etc. etc.
Paraphrasing Archimedes, all I need to know is the {\bf Right Ansatz}
and my computer will prove the Riemann Hypothesis.

As already mentioned before,
often we are stumped because we don't have enough parameters.
In the present humble case, it was impossible to 
evaluate the  determinant of the 
$n \times n$ Hilbert matrix $(1/(i+j-1))$,
because it only depended on the
single parameter $n$, but evaluating the more general
$\det (1/(r+i+j-1))$ was possible, since
this enabled induction on $n$ {\it and} $r$.

So the reason that my computer is unable, at this time, to prove
the Riemann Hypothesis, is that $\zeta(s)$ only depends on one
variable. With an appropriate generalization, belonging to an
appropriate ansatz, it would be doable.

{\bf Building Up Experiments Is Even more important than Performing Them}

Physical science consists of an interplay of {\it theory}
(hypothesis-forming) and {\it experiments} (hypothesis testing).
But in order to perform experiments, one has to {\it design} them,
and {\it build instruments}. So ultimately, the most important
members of the scientific community are neither theorists nor
experimentalists, but {\it engineers} and {\it technicians}, who
build the instruments needed to carry out the experiments.

The analog of scientific instruments are {\it software}.
My role in this project was neither {\it hypothesizing},
i.e. conjecturing, nor, {\it experimenting}, it was all done by
my computer. All I did was 
`build the equipment', i.e. design an algorithm 
and implement it, that is, write the program. 
But this would not have been possible without the computer algebra
system, and its associated {\it programming language},
(Maple in my case), developed through many years 
of dedicated labor by such pioneers
as Keith Geddes and Gaston Gonnet. And of course, even more
importantly, the meta-equipment, i.e. the computer itself,
that is the {\it hardware}.

Eventually, such computer programs will also be written by computers,
thanks to meta-programs and 
future meta-Maple. But hopefully there will still be a role
for humans, in thinking up new ansatzes and {\it meta-ansatzes},
and in finding the {\it trivializing generalization}.
But then again, eventually computers will learn how to do these things
all by  themselves, and we might not be able to 
even follow the general {\it drift} of what they are doing,
because of its immense complexity.

{\bf I am Even More Like My Grandfather Paul Than I Thought Before}

My grandfather Paul Alexander was a chemist,
hence an experimental scientist. However he was not a `pure' scientist,
but rather an `applied' and `industrial' one, 
who designed {\it verfahren}s to do
specific tasks, in his case, recycling rubber. In my case too,
I am not interested in probing the `nature of mathematics' per se,
only in designing {\it algorithms} do to specific tasks.
That specific task happens to be discovering and proving mathematical
facts, but the `recycled rubber' itself 
(i.e. mathematical theorems) are much less exciting than the
process (i.e. computer program) that generated them.

{\bf REFERENCES}

[AE1] Tewodros Amdeberhan and Shalosh B. Ekhad,
{\it A  condensed condensation proof of a determinant evaluation
conjectured by Greg Kuperberg and Jim Propp},
J. Combin. Theory Ser. A 
{\bf 78} (1997), 169-179.

[AE2] Tewodros Amdeberhan and Shalosh B. Ekhad, {\it
Determinants through the looking glass}, Adv. Appl. Math.
{\bf 27} (2001), 225-230.

[D] Charles L. Dodgson, ``{\it Condensation of Determinants}'',
Proceedings of the Royal Society of London {\bf 15} (1866), 150-155.

[H] Wolfgang Hackbusch, {\it From Classical Numerical Mathematics
to Scientific Computing}, Documenta Mathematica, extra volume,
ICM 1998, I, 235-254 (based on a plenary talk).

[M] Percy A. MacMahon, ``{\it Combinatory Analysis}'', vol. 2,
Cambridge University Press, 1918. [Reprinted by Chelsea, 1984].

[Z1] Doron Zeilberger, {\it Reverend Charles to the Aid of
Major Percy and Fields Medalist Enrico},
Amer. Math. Monthly {\bf 103} (1996), 501-502.

[Z2] Doron Zeilberger, {\it Dodgson's determinant evaluation rule
proved by TWO-TIMING MEN and WOMEN}, Electron. J. Comb. {\bf 4(2)}
(1997), R22.

\bye
