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 \line{\SC Rutgers University \hfil\sc New Brunswick
  New Jersey 08903}
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 \line{\sc Department of Mathematics
      \hfil     e-mail: {\tt zeilberg@math.rutgers.edu}}
\line{(732) 445--1326 (voice) \hfil (732) 445--5530 (Fax)}
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%       Date goes here ***********************************
Nov. 25, 2001 \hfil}
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{\obeylines
School of Mathematics
Inst. Advanced Study
Princeton, NJ 08540 
}
 
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Dear Committee,
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This is a letter of recommendation on behalf of Tewodros Amdeberhan, who is
applying for membership at IAS for AY 2002-2003.
Tewodros earned his Ph.D. in 1997, under my direction.
 
Tewodros's thesis was a very important contribution to
{\it algorithmic proof theory of identities}, with elegant
and significant applications to {\it number theory} (Ap\'ery-like
irrationality proofs), and {\it determinant evaluations}.
He has the very rare gift of being equally strong both in
theory and computation. He is the master of the rigorous proof
on the one hand, and at the same time he is a true computer whiz,
who can write long Maple code in order to solve a problem.
 
Tewodros's thesis consists of three loosely related parts. The first part is
in {\it convergence acceleration series}. Using WZ-cohomology,
Tewodros found amazingly fast converging series for
$\zeta(2)$ and $\zeta(3)$, much faster than Ap\'ery's celebrated ones.
One of his formulas:
$$
\zeta(3)=\sum_{n=0}^{\infty}(-1)^n{{n!^{10}(205n^2+250n+77)} \over
{64(2n+1)!^5}}.
$$
was recently used by Simon Plouffe and George Fee
to compute $\zeta(3)$ to more than half a million digit, a world record!
See the Web page on mathematical constants at
 
{\tt http://www.mathsoft.com}. This is also mentioned in
the famous ``Eric's Mathworld'' website:
{\tt http://www.mathworld.com}.
 
The second, perhaps deepest, part of Amdeberhan's thesis is
the irrationality proofs of the $q-$analogs of $\ln(2)$ and
the Harmonic series (which converges when $q>1$). These are
$$
h_q(1):=\sum_{k=1}^{\infty} {{1} \over {(q^k-1)}} 
$$
and 
$$
Ln_q(2):=\sum_{n=1}^{\infty}{ {(-1)^n} \over {q^n-1}}.
$$
respectively.
The special
case $q=2$ was done by Paul Erd\"os in 1947, and the general case
was done recently by Peter Borwein. However, Amderberhan's methods
give the best irrationality measure to date. More importantly,
there is a good chance that they would generalize to irrationality
proofs of the $q-$analogs of $\zeta(2)$ and $\zeta(3)$.
 
The third part is just as impressive. Tewodros proved a
six-year-old conjecture of Greg Kuperberg and Jim Propp.
In fact he did much more. He found a q-analog and a far-reaching
generalization. The importance of exact evaluations of determinants
is now rapidly increasing. I am sure that Tewodros has a very good
chance to employ similar methods to
prove a conjectured determinant evaluation of
Bombieri, van der Poorten, and Hunt, that has far-reaching
number-theoretic consequences. 
 
In addition to Tewodros's mastery of Maple,
he is very computer-literate.
You are welcome to look him up in his own Home Page:
{\tt http://www.math.temple.edu/$\tilde{\quad}$tewodros}.

Since his Ph.D., in spite of a very heavy teaching load,
Tewodros continued to do excellent reseach branching out
into differential equations.

Tewodros is also a gifted and concerned teacher, 
a very kind and responsible human being, who would make
a great colleague. He is very generous with his time, and is always
helping other students with their computer and mathematics problems.
 
To sum up, Tewodros is a very talented and promising
mathematician, with equal strengths in theory, applications, and
computations. He is also a very good teacher and a very fine
human being.  
 
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  \hbox{Sincerely,}\vfil\hbox{Doron Zeilberger}\hbox
{Board of Govs. Professor}}
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