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\bf
\centerline
{
COMPUTERIZED DECONSTRUCTION
}
\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 \quad ,} \hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/   .}
First version: June 21, 2002.  
Expanded version of a plenary talk given at
ECCAD 2002, LaGuardia Community
College of the City University of New York, May 18, 2002, organized by
William Sit and Jerry Ianni.
Accompanied by the Maple packages {\tt PADE, AperyWZ, AperyRecurrence, 
AperyAcc, AperyAppx} 
available from \hfill \break
{\eighttt http://www.math.rutgers.edu/\Tilde zeilberg/apery.html} .
Supported in part by the NSF. 
} 
}
 
{\bf Abstract:} 
The inequality (DERRIDA+TURING)$>$(DERRIDA)+(TURING) will be
illustrated by computerized deconstruction of Roger Ap\'ery's
miraculous proofs of irrationality.


{\bf Preamble}

It is a pleasure to be here, and I thank William Sit
for inviting me and giving me the opportunity
to listen to the fascinating talks by Gert-Martin Gruel 
and Sam Dooley this morning. I also enjoyed the
interesting posters and software demonstrations.

Gert-Martin Gruel started his intriguing talk about SINGULAR
by quoting Sir Michael Atiyah's `provocative'
statement ([At]) that likened the use of Computer Algebra Systems
(and more generally of algebra itself) to a Faustian agreement
whose cost is the sale of the `geometrical intuition' soul.
This made Gert-Martin feel a bit guilty.

While a guru like Atiyah deserves to be taken seriously no matter
what he says,
I disagree with his prejudiced statements whose anthropocentric
tenor reminds me of another (one-time) Oxford don, G.H. Hardy.
Hardy's `apology' used to outrage me, with the artificial and
fictional dichotomy of pure vs. applied, trivial vs. non-trivial,
and  mathematics being a `young {\it man}'s game'. In fact
the hero of today's talk, Roger Ap\'ery, is a great counterexample
to the last statement,
since he was sixty-two when he made the breakthrough that 
was going to immortalize him. 

Because of Sir Michael's immense stature, most people
who disagree with him feel that they have to be polite and
defensive. One beautiful defense of combinatorics, 
and more generally of the `problem-solving culture',
against Atiyah's `theory-building'-supremacy,
was launched by Tim Gowers[G]. I highly recommend it!

But another strategy of rebuttal to elitist and
prejudiced opinions is {\it offense}. So let
me counteract provocation by provocation, and state the
following:

{\bf
Computer Algebra Systems are NOT the Devil but the
new MESSIAH that will take us out of the current utterly
trivial phase of human-made mathematics into the much deeper
semi-trivial computer-generated phase of future mathematics.
Even more important, Computer Algebra Systems will turn out to
be much more than just a `tool', since the methodology 
of computer-assisted and computer-generated research will rule in
the future, and will make past mathematics seem like alchemy
and astrology, or, at best, theology.}

Developers of CASs, and the people who design and implement 
the algorithms, are the unsung heroes of this budding revolution. Even today
CASs are still a {\it marginal} subject (sociologically 
speaking, of course).
For example, it is a scandal that
Bruno Buchberger, whose impact on mathematics, present and future,
is at least as great as Atiyah's, does not get the same recognition
as the latter. But all revolutions take time, and let's hope
that once we become top dogs, we will be kinder and more tolerant
to the future underdogs, including those that will continue to
practice `naked-brain math'.

{\bf Geometrical vs. Combinatorial Intuition}

According to Atiyah, the first half of the 20th century was dominated
by Hilbert (i.e. `formalism' and algebra) while the second half was
dominated by Poincar\'e (`geometrical intuition', the reign of Topology).
Between the lines he patronizes the
mathematicians of the 19th century with their focus on the
{\it local} and on {\it explicit formulas}.

But, as Tim Gowers has recently pointed out to me, 
{\it combinatorial intuition}, that is behind humans'
symbolic manipulation capabilities, is equally important
for progress. Hence it is unfair and untrue to
emphasize geometrical intuition at the expense
of our, at least as important,
combinatorial intuition, and dismiss algebra as
`mindless turning of a crank'.

Ultimately, Hilbert's formalist approach will prove the {\it winner}
(if contest it is).
After all, as was observed by Gregory Chaitin, it lead, via
Turing, to {\it programming},
which is the epitome of formalism. Now Programming 
will survive long after Topology will be forgotten.

Furthermore,
you can only go so far with `geometrical intuition'. Atiyah
himself admits that Topology greatly advanced thanks to physics
(e.g. Yang-Mills, Seiberg-Witten). 
But he seems to be unaware that it is not the `physical intuition'
in the physics, but the combinatorial nature of the physics culture
that was so crucial.
Once  combinatorics will get more advanced
(thanks to computers), it will advance Topology, and everything
else, much more.

{\bf Math and Philo}

Math is {\it perfect} (in principle), but mathematicians are {\it not}
(because they are humans), hence the mathematics that
(human) mathematicians do is influenced by the {\it weltanschauung}
of the people around them. For example the intuitionists were
influenced by {\it phenomenology}, and the Bourbakists by
{\it structuralism}. Not to mention the notorious Nazi mathematicians
who believed in German {\it intuition} as opposed to
`Jewish' {\it formalism} (never mind that Hilbert was a Protestant).
Conversely, mathematics had a profound
influence on philosophy as far back as Pythagoras, through Plato,
Spinoza, Kant, Frege, Wittgenstein, Russel, Husserl,
Heidegger, and almost
everybody, all the
way to Derrida. Speaking of Derrida, Vladimir Tasic has a very
insightful paper[T], soon to be expanded into a book, about
Deconstruction and mathematics.

In this lecture, I will not discuss {\it philosophy} 
per se, but will attempt to show how Derrida's seminal
insights have the potential to revolutionize 
the practice of {\it doing mathematics}.
When interfaced with the computer, of course.

Not all mathematicians and scientists appreciate Jacques Derrida,
and for some, like Alan Sokal, he is a dangerous
enemy of science and `progress', since he, along with
his fellow postmodernists,
seem to undermine the blind faith of most mathematicians
in {\it immutable truths}.
But I am sure that once we get over these hang-ups,
and learn how to deconstruct mathematics, both {\it globally } and
{\it locally}, we will be much better off, and this will enable us
to advance mathematics from its present utterly trivial state
to a much more advanced, semi-trivial, state.

{\bf Global Deconstruction of Math}

Jacques Derrida deconstructed Western metaphysics by challenging
binary opposites like {\it cause and effect}, {\it presence and absence},
{\it speech and writing}, and {\it identity and difference}, with a tacit
dominant concept in every pair.
We should likewise deconstruct the pairs
{\it rigorous vs. non-rigorous} (see [Z2]),
{\it pure vs. applied}, {\it theorem vs. conjecture}, {\it empirical 
vs. theoretical},
and {\it a priori truth vs. experimental truth}, and the closely
related dichotomy {\it deduction vs. induction}.

As argued in [Z2],
the default {\it proof}, in the future, will be non-rigorous, since
we won't be able to afford completely rigorous proofs, except for the
most trivial results, with semi-rigorous proofs a transitional compromise.
The supremacy of {\it pure} mathematics, fortunately, is already
declining, as it is realized that the distinction is only sociological.
Also,  in the future, the lines of demarcation between `theorem' and
`conjecture' will be blurred, and all (non-trivial)
knowledge, even `theoretical', will be empirical and inductive, 
in the sense that it will be all computer-generated.

{\bf Local Deconstruction}

Since so far mathematical proofs are written by humans, they always suffer,
to varying extents, redundancy and hiding the {\it bottom-line idea}
in a smoke-screen of human fluff. Also, sometimes a good proof
starts out combinatorial, but then gets ruined by ``mainstreamers"
who dislike both ``combinatorics" and ``heavy computations".
For example, I LOVE, and really understand, Viggo Brun's 
beautiful and seminal original paper on
the Goldbach problem, and the subsequent
improvements by Buchstab, Wang, Chen, and others. But when I tried
to read a contemporary account I got very depressed.
All the beautiful combinatorial ideas 
got ruined by human verbiage, boring definitions, and 
endless notation.

{\bf
In this lecture, I will propose a methodology of starting out with a human
proof, stripping it of its tacit human over-head and fluff,
then ENCAPSULATING
and formalizing it as  COMBINATORIAL OBJECTS (in the general
sense, in which algebra is part of combinatorics), 
then  PROGRAMMING and trying to let the computer try to find
new objects of the same kind. Often, the formalized object suggests
natural generalizations, that in turn, can be programmed, and this
can continue indefinitely, always getting feedback from the computer's
output.}

This is, in a way, a {\it computerized version} of Tim Gowers's
{\it P\'olya project} (see his website).

Now, when I say {\it proof}, I don't always mean it in the
Euclidean sense of a sequence of statements glued by logical
deduction. Many so-called proofs are really
{\it algorithms} or other {\it objects} (equations,
recurrences, etc.) in disguise, 
and once deconstructed can often be found by computer-search.

I will illustrate this methodology by deconstructing,
{\it in four different ways}, one of my all-time favorites:
Roger Ap\'ery's proof of the irrationality 
of $\zeta(3)$. 

The immediate motivation, of course, is to find irrationality proofs
of other (preferably famous) constants. Although
I haven't succeeded yet, I believe that
with more systematic and extensive computer searches, this will
come to pass.
Even more importantly, the {\it methodology} illustrated here
should be instrumental in solving other major open problems.

While my computer and I 
couldn't find new {\it irrationality proofs}, we {\it did} find lots of
new stuff, notably new {\it accelerating recurrences} 
for certain families of constants defined by slowly-converging
infinite series.

{\bf A Crash Course on Irrational Numbers}

The first crisis in mathematics, about 2500 years ago, was caused
by the discovery that the square-root of $2$ is not a ratio of
integers, and we all know the standard proof. However, the original
proof was better, and in modern notation it says that if
$m>n$ are integers such that $m^2-2n^2=0$ then
so are $m'=2n-m$ and $n'=m-n$. Now this {\it reduction formula}
could have been easily found by computer. This is an example
of an {\it ansatz} ( in this case
linear reduction) that {\it encapsulates} an
approach, and makes it amenable to computer search.

It is an easy exercise to prove that $e$ is irrational
(do it right now!, Hint: consider the partial sums), but
it is not quite as easy, but still not too hard,
to do it for $\pi$. This was first
proved by Lambert about 250 years ago. 
Yet no one has any clue today
how to prove the irrationality of $e+\pi$, $e\pi$ (at least one
of them is, though (why?)), The Euler-Mascheroni
constant $\gamma$, Catalan's constant $C$,
or $\zeta(2k+1)$ for $k >1$. Hence it was very exciting news,
back in 1978, when Roger Ap\'ery proved that 
$\zeta(3)$ is irrational.

The general approach used by Ap\'ery , that probably goes back
to Euler, for proving the irrationality of a given constant
$\alpha$ is to {\it construct} (explicitly, or recursively,
or at any rate, effectively) a {\it sequence} of rational
numbers $a_n/b_n$ (where $a_n$ and $b_n$ are integers), such
that $a_n/b_n \rightarrow \alpha$ `fast enough', more
precisely, such that there exists $\delta >0$ and a constant $C >0$
such that for all $n >0$,
$$
0<\vert {{a_n} \over {b_n}} - \alpha \vert <
{{C} \over {b_n^{1+\delta}}} \quad.
\eqno(APPX)
$$
To deduce irrationality, assume that $\alpha=c/d$, then
$$
\vert {{a_n} \over {b_n}} - \alpha \vert =
\vert {{a_n} \over {b_n}} - {{c} \over {d}} \vert =
\vert {{a_n d - b_n c} \over { d b_n} } \vert >{(1/d) \over {b_n}},
$$
a contradiction.

If the  sequence 
$b_n$ is of exponential growth, then it is easy to see that
the existence of such a $\delta$ implies irrationality measure
$1+1/\delta$ (see [vdP]), so one game people play is to try to lower
the world's record, by constructing better and better approximating
sequences that decrease the known upper bound for the irrationality
measure of the studied constant.

Note that whenever $b_n$ is of exponential growth,
say $b_n=O(A^n)$ for some $A>1$, then the error in the
approximation $(APPX)$ is $O(1/(A^{(1+\delta)})^n)$, hence
is of interest from a numerical-analysis point of view
even when $\delta<0$, provided that  $\delta >-1$.
If $\delta$ is less than $0$, but, close to it,
there is always hope that some accelerated variant of the
approximating sequence will make it. Hence, while it is
a major breakthrough to find an approximating sequence
for a famous constant with $\delta>0$, it is still of interest
to get $\delta$ as big as possible, even if it is negative.

{\bf The Continued Fraction Ansatz}

There is a well-known, and extremely simple, algorithm to
construct a sequence of rational approximations, $a_n/b_n$,
for any given constant (let's call it $\alpha$),
with the impressive $\delta=1$! 
The algorithm is called {\it continued fraction conversion}, 
and the $a_n/b_n$ are the so-called {\it convergents}.

In Maple one types:

{\tt convert(alpha, confrac, a) ; }

and then typing {\tt a;} will give you the sequence of convergents
up to the precision implied by {\tt Digits}. Setting {\tt Digits}
higher and higher, will get you further and further.

There is only one {\it slight problem} with this beautiful idea. The sequence
$\{a_n/b_n\}$ has to be an {\it infinite} sequence, in other words,
the continued-fraction expansion must be {\it non-terminating},
in other words, the constant $\alpha$ must be {\it irrational}.
But that is exactly what we are trying to prove!, so this
is circular reasoning.

But, if we (or the computer) can {\it guess} a {\it pattern},
belonging to an explicit {\it ansatz}, then we can
define $\alpha'$ to be the constant that is given by the infinite
(simple) continued fraction obeying that pattern. Then we can prove
(either manually or, preferably, automatically) that $\alpha'=\alpha$, and
since $\alpha'$ is {\it a priori} irrational (being given by
an effective {\it infinite} simple continued fraction), we have a proof that
$\alpha$ is indeed irrational.

To emphasize the simplicity of this approach, let me not use
the built-in Maple continued-fraction conversion command, but
a home-made one, {\tt CF(a,k)}, that inputs a constant
{\tt a} and an integer {\tt k}, and outputs the first {\tt k} terms in its
continued-fraction representation.

{\tt CF:=proc(a,k) local n: Digits:= 100: \hfill\break
if k=0 then RETURN([]) else 
 n:=trunc(evalf(a)):RETURN([n,op(CF(1/(a-n),k-1)]): 
fi: end:
}

Now the fun begins!

Typing: {\tt CF(sqrt(2),20);} would immediately return:
{\tt [1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]}. 
You don't have to be a genius to conjecture that
the continued fraction of $\sqrt{2}$ is 
$[1,2^\infty]$. Calling this latter number $\alpha'$,
we see immediately that $\alpha'=1+1/\alpha''$ where
$\alpha''=[2^\infty]$, which implies that
$\alpha''=2+1/\alpha''$, which yields $\alpha''=1+\sqrt{2}$,
and hence $\alpha'=\sqrt{2}$, QED.

I did the above example in great detail in order to illustrate
that it can be easily mechanized. The computer can be trivially
programmed to 
do the following steps:
(i) detect the ultimate period of the sequence
(of course of bounded length) , (ii) define the
conjectured infinite continued fraction,
(iii) find automatically the algebraic equation it satisfies, and
finally (iv) identify it with the input constant, 
that in this case has to be a solution of a quadratic equation.

Of course, this is but 
an extremely simple toy example. We already know, thanks
to Legendre, that a continued fraction is
ultimately periodic if and only if it is a quadratic irrationality,
and we also know the classical
and easy result, that algebraic numbers whose minimal equation have
degree higher than one are irrational. But, if we did not know that,
then the above procedure could be used to proved the irrationality of
$\sqrt{5}$, $\sqrt{7}$, etc., and at the same time establish
their optimal irrationality measure, which is $2$.

Another example is {\tt CF(sqrt(3),20);} that yields

{\tt [1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1]},

and we are immediately lead to conjecture that
$\sqrt{3}=[1,(1,2)^{\infty}]$, and once conjectured, it is 
trivial (and purely mechanical) to prove.

A more interesting pattern emerges with {\tt CF(exp(1),20);} that yields

{\tt [2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1] } \quad,

for which it is also easy to detect a pattern. It is slightly harder
to prove from scratch that the infinite continued fraction
of $e$ is indeed what the above suggests (one has to find a more
general statement, also by computer experimentation, and then
use WZ theory).

Consider instead the better looking:

{\tt CF((exp(1)-1)/(exp(1)+1),20);} that yields

{\tt [0,2,6,10,14,18,22,26,30,34,38,42,46,50,54,58,62,66,70,74]} .

Just as easy is 

{\tt CF((exp(1/2)-1)/(exp(1/2)+1),20);} that yields

{\tt [0,4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124,132,140,148]} ,

and

{\tt CF((exp(1/3)-1)/(exp(1/3)+1),20);} that yields

{\tt [0,6,18,30,42,54,66,78,90,102,114,126,138,150,162,174,186,198,210,222]} .

From which the human (or machine!)
can easily guess that the continued-fraction
of $(exp(1/k)-1)/(exp(1/k)+1)$ is

$[0,2k,6k,10k,14k, \dots , (2+4i)k, \dots ]$.

Now this {\it more general} statement can easily be proved, both
by human and machine.

{\bf Empirical AND Rigorous Math}

The general strategy for using computer experiments in order to
obtain {\it rigorous} results is the following.

{\bf Step 1}: Input $Problem(n)$, parameterized by integer $n$.

{\bf Step 2}: Apply a standard (or new) {\it numerical} algorithm
to get $Answer(n)$ for $n=1,2, ..., 100$ (or whatever).

{\bf Step 3}: Have the human, or much better still, the
{\it machine}, guess, from the output of Step 2, the
{\it symbolic} answer $Answer(n)$, for {\it symbolic} $n$
(being a nominalist, I prefer to talk about $n$ as a {\it letter}
rather than say {\it arbitrary} $n$; the latter reflects
Fortran mentality of thinking of $n$ as a variable standing for
concrete integers). How can a machine guess? Easy. All you have to do
is teach it {\it ansatzes} to do `curve-fitting', e.g. the
Salvy-Zimmermann Maple package {\tt gfun}, or Sloane's
{\tt superseeker} (based on the former, I believe, that in turn
was inspired by the pioneering efforts of Simon Plouffe).
See also my package (available from my website) {\tt SCHUTZENBERGER}.

{\bf Step 4}: Have the machine {\it automatically} prove
the guess of Step 3, by `plugging' $Answer(n)$ into the
algorithm, keeping $n$ as a symbol. This should imply
a certain {\it identity}, that should be automatically provable
provided it belongs to the right {\it ansatz}, for example {\it rational
functions} (since Viete), or the {\it holonomic ansatz}
(since WZ). If it does not fit into a known ansatz (framework)
you can either {\it cheat} and find a {\it human} (possibly
computer-aided) ad-hoc proof, or better still, develop a new
{\it algorithmic proof machine} for a new ansatz that will include
$Answer(n)$ and the needed identity. Also remember the 
{\it P\'olya Principle}, of finding the {\it trivializing generalization},
that is, look for $Answer'(n,m_1,m_2, ..)$ that is algorithmic, and such that
$Answer(n)=Answer'(n,0,0, ..)$.

\vfill\eject
{\bf Another Venerable Algorithm: Pad\'e Approximation}

Recall that if a function $f(x)$ has a Taylor expansion
around $x=0$, then the $(m,n)$ {\it Pad\'e approximant} is
the rational function $P_{m,n}(x)/Q_{m,n}(x)$, where
$P_{m,n}(x),Q_{m,n}(x)$  are polynomials in $x$ 
of degrees $m$ and $n$ respectively (and $Q_{m,n}(0) \neq 0$),
and
$$
f(x)- {{P_{m,n}(x)} \over {Q_{m,n}(x)}}= O(x^{m+n+1}) \quad,
$$
or equivalently,
$$
f(x)Q_{m,n}(x)- P_{m,n}(x)= O(x^{m+n+1}) \quad.
\eqno(Pade)
$$

Now, using {\it undetermined coefficients}, one
can easily write a simple procedure that inputs $f(x)$, and
{\it specific integers} $m$ and $n$, and outputs the 
corresponding $P_{m,n}(x)$ and $Q_{m,n}(x)$. All one has to do is use
basic linear algebra that is built-in in Maple (and
of course in most other systems), via {\tt solve} (or its more
specialized variant {\tt linsolve}).

While Pad\'e approximation is built-in in Maple, it is always a good
idea to write your own version, since then it is easier to
modify and generalize. Hence I wrote
my own home-made package {\tt PADE}, available from the
website of this article. Let's play with $e^x$.

First download {\tt PADE} to your current directory. Then go into
maple by typing {\tt maple} (or {\tt xmaple}, 
or clicking on the Maple icon or whatever is
applicable on your computer). Once in Maple, type
{\tt read PADE;} (depending on the system, you may have
to type-in the full path).
Now follow the on-line help.
The main  procedure is {\tt Pade1}, whose syntax is
{\tt Pade1(f,x,m,n);}.

Inputting {\tt Pade1(exp(x),x,1,1);} yields
$$
-{{2+x} \over {-2+x}} \quad,
$$
while inputting {\tt Pade1(exp(x),x,2,2);} yields
$$
{{12+6x+x^2} \over {12-6x+x^2}} \quad,
$$
and inputting {\tt Pade1(exp(x),x,3,3);} yields
$$
-{{120+60x+12x^2+x^3} \over {-120+60x-12x^2+x^3}} \quad,
$$
and inputting {\tt Pade1(exp(x),x,4,4);} yields
$$
{{1680+840x+180x^2+20x^3+x^4} \over {1680-840x+180x^2-20x^3+x^4}} \quad .
$$
It is obvious already that there must be some ``pattern'', if nothing
else because the coefficients are {\it round}, i.e. products of
small primes, which usually indicates that they are expressible
in terms of factorials. Since the denominator $Q_{n,n}(x)$ seems to
equal $P_{n,n}(-x)$, let's focus on the numerator, $P_{n,n}(x)$.
In order to study the coefficients, let's define {\it on the fly},

{\tt a:=(i,j)->coeff((-1)**i*numer(Pade1(exp(x),x,i,i),x,i-j);}

Now the leading coefficients, for $i=0 \dots 10$, are obtained by typing

{\tt seq(a(i,0),i=0..10);} ,

which returns {\tt 1,1,1,1,1,1,1,1,1,1,1} .

Hence we are safe in conjecturing that $a(i,0)=1$ for {\it all} $i\geq 0$.

Next, let's look at the second-to-leading coefficient, by typing

{\tt seq(a(i,1),i=0..10);}

which returns: {\tt 0,2,6,12,20,30,42,56,72,90,110} .

Any human with IQ $\geq 110$ can guess that
$a(i,1)=i(i+1)$. To be more sure, we can type

{\tt seq(a(i,1)/(i*(i+1),i=1..20);}

and indeed get : {\tt 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} .

Next, let's type: {\tt seq(a(i,2),i=1..10);} .

This looks a bit more complicated, but inspired by the conjectured
explicit expression for $a(i,1)$, let's try

{\tt seq(a(i,2)/(i*(i+1)),i=1..10);}

which returns: {\tt 0,2,5,9,14,20,27,35,44,54} .

This is clearly $(i-1)(i+2)/2$, for $i=1..10$.

This leads us to input:

{\tt seq(a(i,2)/((i-1)*i*(i+1)*(i+2)/2),i=2..20);}

with the output: {\tt 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} .

Even though there is already a ``pattern of patterns''
emerging, in order to be safe, try next:

{\tt seq(a(i,3)/((i-2)*(i-1)*i*(i+1)*(i+2)*(i+3)/6),i=3..20);}

with the outcome

{\tt 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1} .

And finally define

{\tt b:=(i,k)->a(i,k)-(i+k)!/((i-k)!*k!);}, 

and test:

{\tt seq (seq(b(i,k),i=1..12),k=1..10);}

to get the very promising output:

{\tt 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 }.

This leads us to conjecture that:

$$
P_{n,n}(x)=\sum_{k=0}^{n}
{{(n+k)!} \over {(n-k)! k!}}x^{n-k} \quad,
\eqno(PadeNumer)
$$ 
and
$$
Q_{n,n}(x)=\sum_{k=0}^{n}
{{(n+k)!} \over {(n-k)! k!}}(-x)^{n-k} \quad.
\eqno(PadeDenom)
$$ 

But this is all very nice, would the humanist say, yet it is
still just a conjecture, we surely need a human to prove this
conjecture for {\it all} $n$, don't we?

As a matter of fact, we don't! Plugging the {\it symbolic} answers
$(PadeNumer)$ and $(PadeDenom)$, and
$$
e^x= \sum_{k=0}^{\infty} {{x^k} \over {k!}},
$$
into $(Pade)$, and comparing coefficients on both sides, results
in a {\it binomial coefficients identity}, that thanks to
W and Z (and of course, Sister Celine) is now completely
automated. In this very simple example the identity turned
out to be well-known (it is Vandermonde-Chu), but that's a historical
accident.

The next argument of the
human chauvinist would be that perhaps each step
is computer-aided, or even computer-generated, but the above
derivation, as a whole, is human
reasoning {\it par excellence}, the way Euler, Gauss,
Riemann, Ramanujan, and all the other great
(and not-so-great) mathematicians have always worked,
but using paper-and-pencil instead of a computer. 
Hence the only
difference is {\it quantitative}, where the computer fills-in
the details a {\it bit} faster. Hence (would the humanist continue),
this is not a true {\it paradigm shift}, but merely
one of convenience.
Recall George Andrews's famous line:
``a computer is {\it nothing but} a pencil with power-steering''.

To this {\it wicked son} thou shall reply: all the above
steps, of guessing, and meta-guessing, and formulating the
general conjecture, and proving, can be combined automatically,
and then it would be done much faster than the above
`interactive' mode. It is true that, 
since at present the programming
is still done by humans, it may be a good idea to first get a feel
for it by playing interactively, but NOT for proving `new results',
that's a waste of time! The only purpose is to inspire us to write
a good guessing/proving program, that ultimately can explore
many more ansatzes, and look for much more complicated patterns,
than any human can do, even by interacting with a computer.

To get an irrationality proof for $e=exp(1)$, plug-in
$x=1$. $P_{n,n}(1)$ and $Q_{n,n}(1)$ are obviously
integers, and the Zeilberger algorithm gives a three-term recurrence
that easily implies
the well-knwon fact that the {\it irrationality measure}
of $e$ is {\it exactly} $2$.

{\bf Ap\'ery's Miraculous Proof}

Ap\'ery's original proof, as described in his notorious
June 1978 Marseille-Luminy talk (see [vdP]) was a sketch
consisting of {\it unlikely assertions}. The details were
worked out by Henri Cohen and Don Zagier, and were beautifully
described in van der Poorten's lively exposition.

Here is a summary:

{\bf Step 1}: The sequence of partial sums of $\zeta(3)$,
$$
c(n):=\sum_{m=1}^n {{1} \over {m^3}} \quad ,
$$
converge extremely slowly to $\zeta(3)$. Hence, let's pull out
of the hat the following double sequence
$$
c(n,k):=\sum_{m=1}^n {{1} \over {m^3}} 
+\sum_{m=1}^{k} {{(-1)^{m-1}} \over {2 m^3 {{n} \choose {m}}
{{n+m} \choose {m}}}}
\quad ,
$$
that has the property that $c(n,k) \rightarrow \zeta(3)$ as
$n \rightarrow \infty$, uniformly in $k$. In particular,
$\vert \zeta(3)-c(n,n) \vert $ is exponentially small.

{\bf Step 2}: The drawback of the $c(n,n)$ is that they seem to
have a huge denominator, so the implied (experimental) $\delta$ in
very negative. On the other hand the denominators of the
$c(n)=c(n,0)$ are not too bad, they are $lcm(1,2, \dots, n)^3=O(e^{3n})$.
So let's try to form a {\it weighted average},
for a {\it judicious} choice of weights $b(n,k)$,
$$
x_n:=
{ {\sum_{k=0}^{n} b(n,k)c(n,k)}
\over
 {\sum_{k=0}^{n} b(n,k)}} \quad .
$$
Now, let's pull out of the hat the choice 
$$
b(n,k)={{n} \choose {k}}^2{{n+k} \choose {k}}^2 \quad,
$$
and {\it Voil\'a!}, modulo checking out the details,
the sequence of rational numbers $x_n$ has the desired property:
$\vert \zeta(3) - x_n \vert < C/denom(x_n)^{1+\delta}$
for $\delta=.08\dots>0$.

But how in the world did Ap\'ery come up with this miraculous
$c(n,k)$ and $b(n,k)$? In [Z1] (section 13), I 
placed Ap\'ery's proofs in the context of WZ theory, and indeed
succeeded in mechanizing all the steps but one
(the very easy human step of proving that 
$2 c(n,k) {{n+k} \choose {k}}$ has denominator equal to
$lcm(1,2, \dots, n)^3$). 
All this is now streamlined and generalized
in the new Maple package {\tt AperyWZ}.
I also wrote general procedures that input
arbitrary $WZ$ pairs and arbitrary (closed-form) weights
$b(n,k)$, trying to find the corresponding $\delta$ for
the new constants that emerge. 

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

Download {\tt AperyWZ} from the website of this article

{\tt http://www.math.rutgers.edu/\Tilde zeilberg/apery.html},

and also download {\tt EKHAD}, and put them in the same directory.
To get on-line help for {\tt AperyWZ} type {\tt Ezra();}
({\tt ezra();} is for help with {\tt EKHAD}). To get
help with a specific procedure, type {\tt Ezra(procedure\_name);} .
One of the important procedures is
{\tt zeilWZPv}, and to get help with it type:
{\tt Ezra(zeilWZPv);} (its terse version is
simply {\tt zeilWZP}).

As suggested there, try:

{\tt zeilWZPv(binomial(n,k)*binomial(n+k,k),WZlog2,k,n,N) ;}

{\tt zeilWZPv(binomial(n,k)**2*binomial(n+k,k),WZzeta2,k,n,N);}

{\tt zeilWZPv(binomial(n,k)**2*binomial(n+k,k)**2,WZzeta3,k,n,N) ;}

to automatically perform all the non-trivial steps in 
Ap\'ery's proofs of the irrationality
proofs of $\log 2$, $\zeta(2)$ and $\zeta(3)$ respectively.

{\tt WZlog2, WZzeta2, WZzeta3} are the WZ-pairs that give rise
to the miraculous $c(n,k)$ (as potential function) constructed by
Ap\'ery for the irrationality proofs (see [Z1]).

You should also check out {\tt Acc}, that automatically produces
(rigorously!) the famous accelerated sums that are spin-offs
of Ap\'ery's proofs, and {\it here} one can easily get lots of
new results. Look also at {\tt Constant1}. See the sample input
and output files in

{\tt http://www.math.rutgers.edu/\Tilde zeilberg/apery.html} .

{\bf How to get new WZ-pairs?}

One gold mine of WZ-pairs are explicit well-known hypergeometric
identities. The WZ-pairs that feature in
the irrationality proofs of $\log 2, \zeta(2)$ and $\zeta(3)$ 
turn out to be 
specializations of the 
WZ-pairs that come from the Vandermonde-Chu, Kummer, and Dixon
identities, respectively.

The procedures {\tt WZchu, WZkummer, WZdixon} allow one to 
construct other WZ pairs.

{\bf Starting Out from Binomial Coefficients Sums}

Sometimes it pays to {\it rewrite} history, and not worry so much 
how brilliant proofs were {\it actually} made, but how they
{\it could have been made} or perhaps {\it should have been made}.

So forget about $\zeta(3), \zeta(2), \log 2$ or any other
{\it specific} constant, and let's take as our
starting point a binomial coefficient sum
$$
b(n):=\sum_{k=0}^{n} b(n,k) \quad,
$$
where $b(n,k)$ is a {\it hypergeometric} term. Thanks to the
so-called {\it Zeilberger algorithm} and {\it WZ theory}, it satisfies
some homogeneous linear recurrence equation with {\it polynomial coefficients},
with the obvious initial values $b(0)=b(0, 0), b(1)=b(1,0)+b(1,1) , \dots$.
Now define an {\it associated  sequence} $a(n)$, as the solution of
the same recurrence with other initial values,
(e.g. if ORDER=2 then take $a(0)=1, a(1)=0$). Then, if all goes well
(and often it does), $a(n)/b(n)$ converges to some (as yet unknown)
number $\alpha$, and since the convergence is fast, we can determine
$\alpha$ to high precision.

Then we go to Borweins' {\tt RevENG} website, or send E-mail
to Simon Plouffe,
and check if the constant $\alpha$ is well-known,
and {\it not yet proved to be irrational}.
And if things go {\it really well} it would be (conjecturally for now).
Then you go back and find, still empirically a $\delta$ such that
$|a(n)/b(n)-\alpha|<C/(denom(a(n)/b(n))^{1+\delta}$.
If things go {\it amazingly well} then $\delta>0$, and you would 
almost have a proof that $\alpha$ is irrational. Of course, you would
still have to prove rigorously that your $\alpha$ is the same as
the one suggested by {\tt RevENG}, and that indeed $\delta >0$.
But this shouldn't be too hard, with such a high motivation.

This is implemented in the Maple package {\tt AperyRecurrence}.

{\bf  The Maple package AperyRecurrence}

After downloading it, make sure that you also have
{\tt EKHAD} in the same directory. Now go into Maple
and type: {\tt read AperyRecurrence;}, followed by
{\tt Ezra();}.

The main procedure is {\tt Roger}.
For example, to reproduce the Ap\'ery miracles type:

{\tt Roger(binomial(n,k)*binomial(n+k,k),k,n,20, 40);}

{\tt Roger(binomial(n,k)**2*binomial(n+k,k),k,n,20, 40);}

{\tt Roger(binomial(n,k)**2*binomial(n+k,k)**2,k,n,20, 40);}

since the second component, $\delta$, is positive, this means
that there probably is an irrationality proof behind the
approximating sequence. Since the first components are
`famous' ($\log 2, \zeta(2), \zeta(3)$ respectively), there
would have been reason to rejoice,
had it been discovered this way. See the sample input
and output files in the webpage mentioned above.

It would be interesting to conduct a systematic search, coupled with
the {\tt RevENG } machine.

{\bf Ap\'ery's ORIGINAL Proof}

It is hard to believe that any {\it human} (or even computer)
can pull out of the hat the `winning' $c(n,k)$ and $b(n,k)$
that produced the irrationality proofs, even by WZ-hindsight. 
Frits Beukers [Be]
was able to dispel some of the magic, and do away with recurrences,
and his interesting approach deserves its own computerized deconstruction.
Beukers's approach was extended by Hata[Ha], 
and Rhin and Viola, who hold the
current record for the irrationality measure of $\zeta(3)$.

But the mystery still remained: {\it How in the world did Ap\'ery come up
with his marvelous proof?}.

Fortunately for posterity, Roger Ap\'ery
gave away his secret, in a little-cited,
somewhat sketchy, but {\it gorgeous} paper [Ap],
entitled {\it ``Interpolation de fractions continues et irrationalit\'e
de certaine constantes''}
published in 1981 by the French National Library in their 
{\it Bulletin de la section des sciences du C.T.H.S. \#3
p. 37-53}. One should also mention the exposition and elaboration
of Ap\'ery's ideas (delivered in a Bordeaux
talk) by C. Batut and M. Olivier
[BO], that  was an excellent {\it human deconstruction}.
But for our purposes it is best to {\it read the master} rather
then the disciples, and I will now follow 
Ap\'ery's account in [Ap] very closely.

{\bf A Human Description of
Ap\'ery's Brilliant Acceleration-Convergence Approach}

The terms of the sequence of partial sums of $\zeta(3)$,
$$
{{a(n)} \over {b(n)}}:=\sum_{i=1}^{n} {{1} \over {i^3}} \quad ,
$$
have relatively `small' denominators ($lcm(1, \dots, n)^3=O(e^{3n}))$
but {\it very} slow convergence.

Let's write 
$$
b(n)=n!^3 \quad , \quad 
a(n):=n!^3( \sum_{i=1}^{n} {{1} \over {i^3}} )
$$
Then we have
$$
b(n+1)-(n+1)^3 b(n) =0 \quad , \quad \quad a(n+1)-(n+1)^3 a(n) =b(n) \quad .
$$

Introducing the shift operator $N$, where,
for any sequence $f(n)$, $N^rf(n):=f(n+r)$, the above can be written
as
$$
(N-(n+1)^3)b=0 \quad, \quad (N-(n+1)^3)a=b \quad ,
$$
and applying $(N-(n+1)^3)$ to both sides yields
$$
(N-(n+1)^3)^2b=0 \quad , \quad  (N-(n+1)^3)^2 a=0 \quad .
$$
Expanding gives:
$$
b(n+2)-((n+2)^3+(n+1)^3)b(n+1)+(n+2)^3(n+1)^3b(n)=0 \quad,
$$
$$
a(n+2)-((n+2)^3+(n+1)^3)a(n+1)+(n+2)^3(n+1)^3a(n)=0 \quad.
$$
Hence $\zeta(3)$ may be {\it defined} as
$\lim_{n \rightarrow \infty} a(n)/b(n)$ where
$a(n)$ and $b(n)$ are both solutions of the {\it same}
recurrence, namely
$$
x(n+2)-((n+2)^3+(n+1)^3)x(n+1)+(n+2)^3(n+1)^3x(n)=0 \quad ,
$$
but with different initial values:
$b(0)=1,b(1)=1$, and $a(0)=0,a(1)=1$.

Now we can {\it forget} about the original definition of
$\zeta(3)$ (as an infinite series)
and define it, uniquely, by the three-tuple
$$
[ ope(N,n),[a0,a1],[b0,b1] ] \quad.
$$

Now let $P(n)$ be {\it any} discrete function,
and perform the transformation 
$$
x(n) \rightarrow P(n) x(n)+x(n+1) \quad, 
$$
Defining $a'(n)=P(n)a(n)+a(n+1) \quad $, 
$\quad b'(n)=P(n)b(n)+b(n+1)$,  we have 
$$
\lim_{n \rightarrow \infty} a'(n)/b'(n) =
\lim_{n \rightarrow \infty} a(n)/b(n) \quad .
$$
But Ap\'ery is trying to look for a {\it judicious} choice of $P(n)$ 
such that the rate of convergence of 
$$
\lim_{n \rightarrow \infty} a'(n)/b'(n)
$$ 
to the 
desired constant, $\zeta(3)$,  is {\it faster} then the 
rate of convergence of the {\it original}
$$
\lim_{n \rightarrow \infty} a(n)/b(n) \quad .
$$
Using elementary linear algebra (that is easily
programmed in Maple), one can get ({\it automatically}) 
the 2nd order recurrence operator $Ope_1(N,n)$ annihilating
$x'(n)=P(n)x(n)+x(n+1)$. 

So far $P(n)$ could have been anything, but if one wishes
to stay in the {\it holonomic ansatz} , we have to restrict
attention to {\it polynomial} $P(n)$. Writing $P(n)$ in
generic form, in increasing powers of n, 
$$
P(n)=c_0+c_1n+c_2n^2+\dots + c_d n^d \quad,
$$
(where $d$ is the guessed degree of $P(n)$), and plugging it in,
we get the operator $Ope_1(N,n)$ whose coefficients
of $N^0,N^1,N^2$
are {\it polynomials} in $n$ and {\it quadratic} in the
(yet) undetermined coefficients $c_0,c_1, \dots, c_d$.
It turns out that for best convergence, the coeff. of
$N^2$ in $Ope_1(N,n)$ should be of degree $0$ in $n$.
Hence to find the {\it good} $c_0, c_1, \dots, c_d$,
we (or the computer) have to solve a system of
{\it non-linear} (in fact quadratic) equations in the
unknown coefficients $c_0, c_1, \dots, c_d$.
There is no {\it a priori} guarantee that there is
a solution, or that the solution would consist of
{\it rational numbers}.
But, as Ap\'ery put it, {\it si on a de la chance}, 
things work out.

Now we have a new {\it operator description} of 
the desired
constant, 
$$
[[Ope_1(N,n),[a0',a1'],[b0',b1']] \quad ,
$$
where the coefficient of $N^2$ in $Ope_1(N,n)$ is a constant
(or in general, of the smallest possible degree (in $n$) as possible),
and $a0'=P(0)a0+a1, a1'=P(1)a1+a(2),b0'=P(0)b0+b1, b1'=P(1)b1+b(2)$.

Now let's try and repeat this process indefinitely.
Let's call the previous
$P(n)$, $P_0(n)$. Then we next get (hopefully) 
a good $P_1(n)$ that turns the already better $Ope_1(N,n)$ into
something better still,
$Ope_2(N,n)$. If all goes well, we can keep going indefinitely,
getting a sequence of 2nd-order recurrence operators
$\{Ope_m(N,n)\}$ and accelerating polynomials $P_m(n)$.

Now the {\it second miracle} that happened to Ap\'ery
(for $\log 2$, $\zeta(2)$, and $\zeta(3)$) was that
not only did the increasingly better $Ope_m(N,n)$,
and the accompanying $P_m(n)$, seem to exist for all $m$,
but something even more amazing happened.

It turned out that $Ope_m(N,n)$ and $P_m(n)$
are also polynomials in the {\it index} $m$!
Now if $Ope_m(n,N)$ and $P_m(n)$ are indeed polynomials
in $m$, then it is easy for the computer to guess them, and
we can write
$Ope_m(n,N)=OPE(m,n,N)$ and $P_m(n)=\P(m,n)$, then
plugging the {\it symbolic} $\P(m,n)$ into the accelerating
procedure described above with $OPE(m,n,N)$ as input,
should give the output $OPE(m+1,n,N)$, and this would be
a rigorous proof of the scheme.

Calling the solutions of the `$m^{th}$-row',
$a(m,n)$ and $b(m,n)$, we now have a two-dimensional
scheme such that for each $m$,
$$
\lim_{n \rightarrow \infty} {{a(m,n)} \over {b(m,n)}} = \zeta(3)\ \quad ,
$$
for every $m \geq 0$, where the convergence rate gets better
and better as $m$ increases.

The next miracle was that $b(m,n)$ was not only
divisible by $n!^3$ (as was to be expected) but by
$m!^3n!^3$. Finally $\{A_n/B_n\}$ where
$B_n:=b(n,n)/n!^6$
and $A_n:=a(n,n)/n!^6$, turned out to be
the winning sequence that converges to $\zeta(3)$ fast
enough to get a positive $\delta$.

Note that the acceleration-improvements from one row
to the next is only polynomial, i.e. the $\delta$ for
each individual row is still the worst-possible $\delta=-1$.
But, like for Cantor and Turing, the {\it diagonal saved the day}.

With this (original) Ap\'ery approach, it is not
necessary to use the Zeilberger algorithm,
one can automatically get the recurrence satisfied by both
$a(n,n)$ and $b(n,n)$ (and hence
by $A_n$ and $B_n$), automatically using elementary
linear algebra as follows.

The two-dimensional scheme
is uniquely defined by
$x(0,n)=x_0(n)$ and
$$
x(m+1,n)=\P(m,n) x(m,n)+x(m,n+1) \quad .
$$
In operator notation we have that $x(i,n)=a(i,n)$ and
$x(i,n)=b(i,n)$ are annihilated by the
operator $M-I-\P(m,n)N$ (where $M$ is the shift operator
in $m$: $Mx(m,n):=x(m+1,n)$), as well as
$OPE(m,n,N)$. Now using linear-algebra (that has been
programmed into {\tt AperyAcc} described below) we
(or rather the computer) can easily find a 
{\it diagonal recurrence}  of the form 
$OPER(m,n,MN)$ annihilating both $a(m,n)$ and $b(m,n)$,
and hence an ordinary recurrence operator
$OPER(n,N)$ annihilating
both $a(n,n)$ and $b(n,n)$, and finally one
annihilating both $A_n$ and $B_n$.

{\bf Complete Automation of Ap\'ery's Heuristic Method}

Now that we have the general approach (thanks to the genius of
Ap\'ery), we can program {\it all} the steps, including the
guessing ones, and streamline them, letting the computer
do everything, and outputting {\it failure} if it does not
work out (unfortunately, it does not work for $\zeta(4)$
or $\zeta(5)$, at least not directly).

{\bf A Slightly More Global Approach}

In a way, the {\it input} was the (ordinary) recurrence operator
$ope(n,N)$ (and the initial conditions, but these are not
relevant), and the {\it outputs} were the polynomial in
{\it two} variables $\P(m,n)$ and the operator
$OPE(m,n,N)$. We could find these outputs {\it directly},
if they exist, by writing $OPE(m,n,N)$ and $\P(m,n)$
generically, letting Maple automatically generate the
set of equations implied by $OPE(0,n,N)=ope(n,N)$
and the fact that if
$x'(n)=\P(m,n) x(n)+x(n+1)$, where $x(n)$ is a solution of
$OPE(m,n,N)x(n)=0$ then $x'(n)$ is annihilated
by $OPE(m+1,n,N)$  (this is all done automatically, we
never have to see the resulting equations in the 
guessed coefficients). Then Maple can directly find
the miraculous $\P(m,n)$, without the intermediate
guessing. Of course, solving the system of equations
is just implicit guessing, so the difference between
the two approaches is not fundamental.

The advantage of the latter approach is that once we find
the magic $\P(m,n)$ (and its associated $OPE(m,n,N)$),
by doing undetermined coefficients, we don't have to
do the final verification stage (because that's how we
found it in the first-place). In the previous, piecemeal
method, the initial $\P(m,n)$ was but a {\it conjecture},
and one had to verify it. (Not that it is a big deal,
it takes a few seconds).

{\bf The Maple Package AperyAcc}

The downloading is analogous to the previous packages.
The main procedures are {\tt Aperyh} (or 
if you want the output in operator
notation, {\tt AperyhOper}), and
{\tt AccRec, AperyNes, AperySeq, Appx}, and {\tt RatImp}.
We refer the readers to the on-line help (invoked
by typing {\tt ezra();} then {\tt ezra(procedure\_name)};).
Sample inputs and outputs can be viewed in the
webpage of this article \hfill \break
{\tt http://www.math.rutgers.edu/\Tilde zeilberg/apery.html}.

Even though I was unable to find new irrationalities,
{\tt AccRec} does give amazing recurrences that converge
exponentially fast to constants defined by slowly converging
series. For example
{\tt AccRec(-(2*n-1)**2,-1,n,N); } finds
in a few seconds, and {\it completely automatically},
the recurrence for the fast computation of Catalan's
constant, found, by semi-human means by Zudilin[Zu].

{\bf The Trivializing Hindsight}

I believe that the procedure explained above (based on Ap\'ery's
own honest account [Ap]) is much more
natural and motivated than the official 
version presented in [vdP], that is the epitome
of ad-hocness, and also more natural than Beukers's 
elegant version.

But a close scrutiny of Ap\'ery's approach, as implemented
in {\tt AperyAcc}, leads to {\it yet a simpler} approach, at least
for the {\it discovery} of the proof.
Once you have the {\it magic sequence},
filling-in the details is a comparably minor task.

Recall that initially
$$
b(0,n)=n!^3 \quad ,
\quad
a(0,n)=n!^3 \sum_{i=1}^{n} {{1} \over {i^3}} \quad.
$$

By repeatedly using $a(m+1,n)=\P(m,n)a(m,n)+a(m,n+1)$, and
$b(m+1,n)=\P(m,n)b(m,n)+b(m,n+1)$, we can write
$$
b(m,n)=n!^3 B(m,n) \quad ,
\quad
a(m,n)=n!^3 B(m,n)\sum_{i=1}^{n} {{1} \over {i^3}} 
+n!^3C(m,n)\quad,
$$
for some polynomials in $n$, $B(m,n),C(m,n)$, for
$m=0,1,2, \dots$. These can be computed automatically for each specific
$m$, but in general there is no guarantee
that there is an explicit expression for it as a function of $m$ and $n$
(although in Ap\'ery's case it was true).

Hence, we have that the approximants coming from the
$m^{th}$-row
$$
{ {a(m,n)} \over {b(m,n)}}=
\sum_{i=1}^{n} {{1} \over {i^3}} +
{{C(m,n)}\over {B(m,n)}} \quad,
$$
where the degree, in $n$, of $B(m,n)$, is $2m$,
and the degree of $C(m,n)$ is $2m-2$.
Writing 
$$
R_m(n)={{C(m,n)}\over {B(m,n)}} \quad ,
$$ 
we see that
the rational function $R_m(n)$ is such that
$$
\sum_{i=1}^{n} {{1} \over {i^3}} +R_m(n) \approx \zeta(3) \quad,
\eqno(Keruv)
$$
for large $n$. This means that 
$R_m(n)$ is a `rational-function improvement' added
to the partial sums of the defining series of $\zeta(3)$
in order to get it to be much closer to $\zeta(3)$.

Plugging in $n=n-1$ in $(Keruv)$ and subtracting from
the original yields
$$
 {{1} \over {n^3}} +R_m(n)-R_m(n-1) \approx 0 \quad,
$$
and indeed, the left-hand side turns out to be
a rational function with denominator of degree $4m+3$
and {\it numerator} of degree $0$!. Of course, it would have
been nice if we could have replaced the $\approx$ by $=$ above,
but then $\sum_{i=1}^{n} 1/i^3$ would have been Gosperable
(i.e. Indefinitely summable in closed-form), which would have meant
that $\zeta(3)$ was rational, so it is just as well...

So, by hindsight, this is the most straightforward approach!
Let's generalize it
to trying to find irrationality proofs (or at least,
good diophantine approximations) for constants given
by infinite series of the form
$$
\alpha:=\sum_{i=1}^{\infty} {{1} \over {pol(i)}} \quad,
$$
where $pol(i)$ a polynomial in $i$.

Define the {\it best partial-sum rational improvement} of degree
$m$, to be that rational function, let's call it again
$R_m(n)$, of denominator of degree $m$ in $n$ and
numerator of degree $m-degree(pol,n)+1$ such that
the rational function
$$
 {{1} \over {pol(n)}} +R_m(n)-R_m(n-1)  \quad,
\eqno(Pashut)
$$
has numerator of least possible degree. It is very
easy to write a procedure for finding such $R_m(n)$,
for any specific positive integer $m$,
by expressing the given rational function in generic form,
plugging into $(Pashut)$, and equating to $0$ the coefficients
of the positive powers of $n$ of the numerator of the 
resulting rational function, solving, and plugging back.
Once again, Maple has to solve a system of non-linear equations
with quite a few unknowns, but surprisingly it does it very well,
and not only that, the coefficients turn out to be rational numbers
(at least in the simple Ap\'ery cases and all the other cases
that I have tried).

Once the computer cranked out $R_m(n)$ for, say, $0 \leq m \leq 30$,
we look at the sequence of rational numbers
$$
c_n:=\sum_{i=1}^{n} {{1} \over {pol(i)}}+R_n(n) \quad ,
$$
and, {\it empirically} estimate the $\delta$. If $\delta>0$
then we should open a bottle of Champagne, and
then complete the proof
by any means (human, computer, or combination thereof).
If the empirical $\delta$ is negative, don't be sad!
Better luck next time!

It is true that using this simple approach, even for
$pol(i)=i^2$ (for $\zeta(2)$) and $pol(i)=i^3$
(for $zeta(3)$) only {\it indicates} that a proof is in sight,
rather than gives such a proof. But it is not hard to
complete this approach into a full proof
(if you are feeling lazy, you can always E-mail
Don Zagier or Henri Cohen).

What would have worked in Ap\'ery's cases is the
following.
Once you have the $R_m(n)=B_m(n)/C_m(n)$ you can use
{\tt gfun} or {\tt findrec} to guess recurrences for
$B_m$ and $C_m$ in $n$, and also in $m$, thereby getting,
{\it empirically} for now the $OPE(m,n,N)$ and
$\P(m,n)$ of the previous approach, and once guessed,
it is completely routine to finish-up the proof rigorously
and {\it automatically} using the previous package {\tt AccRec}.
\vfill\eject
{\bf The Maple Package AperyAppx}

All this is implemented in the Maple package
{\tt AperyAppx}, downloadable from this article's
webpage.
The main procedure is {\tt RatAppx}. See the sample input and output
files in the webpage of this article. The package handles the
more general case of improving the acceleration of the
partial sums of convergent series of the form
$$
\alpha:=\sum_{i=1}^{\infty} {{Cf(i)} \over {pol(i)}} \quad,
$$
where $Cf(i)$ is {\it closed form}.

                                    
{\bf Conclusion}

In her excellent book about Derrida, Christina Howells
([Ho], p. 2) describes the Deconstruction strategy
practiced by him as follows. ``In all his books and
essays Derrida is a scrupulous, meticulous, patient
reader, determined to disentangle what has been conflated,
to bring to light what has been concealed, and to pay
scrupulous attention to marginalia and footnotes, in the
expectation that what has been relegated to the margins may
prove paradoxically central to a less parochial understanding
of the text''.

In this lecture, and the accompanying Maple packages, I tried
to apply these principles to {\it human mathematical proofs}
that have been conflated to suit human predilections and
writing styles, and that were meant to be consumed by humans.

By careful analysis of a human proof, one can hopefully
extract the {\it core ideas} and  {\it core objects}, and
then computerize them. Also the {\it marginalia}, in this
case Ap\'ery's little known `expository' and {\it motivational}
paper, proved to be crucial for a successful computerized deconstruction.

This is but a very crude beginning, but I am sure that it is
not the end.

{\bf REFERENCES}

[Ap] Roger Ap\'ery, 
{\it ``Interpolation de fractions continues et irrationalit\'e
de certaine constantes''}
Bulletin de la section des sciences du C.T.H.S. \#3
p. 37-53, 1981.

[At] Sir Michael Atiyah, {\it Mathematics in the 20th Century},
Bull. London Math. Soc. {\bf 34} (2002), 1-15.

[BO] Christian Batut and Michel Olivier,
{\it Sur L'Acc\'eleration de la Convergence de certaines
Fractions Continues}, S\'eminaire de Th\'eorie des Nombres
Ann\'ee 1979-1989- expos\'e no. 23.

[Be] Frits Beukers, {\it A note on the Irrationality of
$\zeta(2)$ and $\zeta(3)$}, 
Bull. London Math. Soc. {\bf 11} (1979), 268-272.

[G] W. Timothy Gowers, {\it The Two Cultures of Mathematics},
in: {\it ``Mathematics: Frontiers and Perspectives''}, edited
by V. Arnold et. al., Amer. Math. Soc., Providence, 2000.

[Ha] Masayoshi Hata, {\it A new irrationality measure for $\zeta(3)$},
Acta Arith. {\bf 92} (2000), 47-57.

[Ho] Christina Howells, {\it ``Derrida''}, Polity Press,
Cambridge, UK, 1998.

[vdP] Alf van der Poorten, {\it A Proof that Euler Missed...
Ap\'ery's proof of the irrationality of $\zeta(3)$},
Math. Intelligencer {\bf 1} (1979), 195-203.

[T] Vladimir Tasic, {\it Poststructuralism and Deconstruction:
A Mathematical History}, available from his website.

[Z1] Doron Zeilberger, {\it Closed Form (pun intended!)}, 
 in: ``Special volume in memory of Emil Grosswald", M. Knopp
and M. Sheingorn,
eds., Contemporary Mathematics {\bf 143} 579-607, AMS, Providence (1993).
 

[Z2] Doron Zeilberger,
{\it Theorems for a price: Tomorrow's semi-rigorous mathematical culture},
Notices of the Amer. Math. Soc. {\bf 40 \# 8}, 978-981 (Oct. 1993).
Reprinted: Math. Intell. {\bf 16}, no. 4, 11-14 (Fall 1994).

[Zu] Wadim Zudilin, {\it Ap\'ery-like Difference Equations
for Catalan's Constant}, preprint, \hfill\break
{\tt arXiv:math.NT/0201024}, available from {\tt xxx.arXiv.org }.
\end 


