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\bf
\centerline{How Sister Celine Fasenmyer 
and Dick Duffin Shaped my Mathematical Personality}
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\it
\centerline{Doron ZEILBERGER\footnote{$^1$}
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\eightrm  \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122.
{\baselineskip=9pt
\eighttt zeilberg@math.temple.edu ; 
\break http://www.math.temple.edu/\Tilde 
zeilberg }.
May 14, 1997.
}}
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\rm
Two of my great heroes passed away last fall. First Richard
James Duffin, at the end of October, and then
Sister Celine Fasenmyer, at the end of December.
Both of them lived to a ripe old age (87 and 90 resp.).
 
Apart from their
longevity,  they seem to have nothing in common. Dick was
recognized as a leading applied mathematician, was a member
of the National Academy of Science, had many brilliant
Ph.D. students, (at least one of whom (Raul Bott) who
surpassed him in fame, and at least another (Hans Weinberger)
of comparable fame), had over two-hundred publications,
etc. etc. On the other hand, Sister Celine Fasenmyer was
an obscure college professor, who did not publish anything
beyond her thesis work, and of course never had any Ph.D.
students.
 
A closer look reveals that they are not as different as
they appear to be at first sight. First, I believe that
they were both grossly under-rated. Sister Celine's
greatness only started to emerge with the WZ theory, and
her story is told in the classic {\it A=B}. I am sure that
the future will prove her even greater.
 
Dick Duffin's
creative genius, while partly recognized, sure did not
get the full recognition it deserved: 
a place among the top $20$ mathematicians of our
century. The main reason
that Dick did not get his proper recognition-due was that
he loved Mathematics more than he liked being famous.
In the pompous era of Bourbakism, he preferred to have
fun and solve problems. Often these problems lead to
new theories, but he never made a big deal of it, continuing
to the next problem. His problem-solving love can be
gleaned from his lovely Bulletin article
({\it Bull. Amer. Math. Soc. {\bf 80}(1974), 1053-1070)},
extended in `Constructive Approaches to Mathematical
Models' (Proc., in honor of R.J. Duffin), C.V. Coffman, and G.J. Fix, eds.,
Academic Press, 1979,pp 3-32.
 
Duffin's Bulletin paper looks like, and in some sense is,
a disjointed collection of `math-bites'. However,
each of these bites is a magnificent contribution, and
the sum of the parts is very considerable. 
A more global view would show that the whole is even more
magnificent than the sum of the parts, and I hope that
in the future, his deep insight, combined with his
down-to-earth unpretentious style
of doing mathematics, will be emulated. I believe that this
paper should be required reading to any budding graduate
student in mathematics, both pure and applied.
 
But, the most important thing that Sister Celine and Dick Duffin
share, is a place in my personal Pantheon of heroes,
who shaped my mathematical personality.
 
According to Sigmund Freud, an individual's personality
is formed in the first five years of his or her life. The mathematical
analog of the formative years is the period of
one's graduate studies. It was then that I encountered,
quite by accident, both Dick Duffin and Sister Celine.
 
My advisor, Harry Dym,
gave me a problem in complex analysis. 
But, as hard as I tried, all the approaches that he
suggested did not seem to work. 
After a year and a half, I was still stumped.
Then I had my own idea: to solve the
problem by doing the discrete analog first. I went to
the library, and started browsing, completely at random,
at current and old math journals.
(Of course MathSciNet did not exist back in 1974, and I was not
systematic enough to look at Math Reviews.)
I then bumped into an article by Duffin and his student
Charles Duris, that had a reference to Dick's seminal
paper on discrete analytic functions, that appeared
in the Duke Journal, in 1956.
 
I immediately fell in love with Discrete Analytic
Functions. I also realized
that I was not born to be a continuous analyst, but
a discrete one. Of course, knowing discrete analytic
functions did not help me a bit with the original thesis
problem, but I did not care, and I duly abandoned it, in favor
of proving discrete analogs of any theorem in complex analysis
I could think of.
 
Neither Harry, nor the other professors, in the then
predominantly analysis math department 
at the Weizmann Institute,
wholly approved of this change of direction. To them it seemed
a futile exercise in `formal analogs', and Yakar Kannai
even pleaded with the great function-theorist, Larry Zalcman,
who was visiting the department, to `talk me out of this
nonsense'. In a sense they were right. Doing
Discrete Analytic Functions, was not the most optimal
way to get an academic job.
 
Later, I also came to realize that as a theory {\it per se},
Discrete Analytic Functions would never have the rich texture
of its continuous namesake. But, studying Dick Duffin's and
his students' papers lead to something far more important.
The {\it realization} that the Continuous and Discrete
should be studied  side by side. Even more significantly,
the {\it operator notation} that Duffin widely
used.
 
It was Duffin who introduced me, through his papers, to this very
powerful notation. 
A {\it discrete analytic function} is a complex-valued
function on the discrete two-dimensional lattice $Z^2$ such that
$$
{{f(m+1,n+1)-f(m,n)} \over {1+i}}=
{{f(m,n+1)-f(m+1,n)} \over {i-1}} \quad .
$$
 
In other words the `directional (discrete)
derivative' in the NE diagonal equals that in the NW diagonal,
at each lattice point. By cross multiplying,
$$
f(m+1,n+1)-i f(m,n+1)+ i f(m+1,n)-f(m,n) \equiv 0 \quad.
$$
 
Duffin defined $Xf(m,n):=f(m+1,n)$ and
$Yf(m,n):=f(m,n+1)$, and rewrote the above as:
$$
(XY-iY+iX-1) f(m,n) \equiv 0 \quad .
$$
 
In other words $f$ is {\it discrete analytic} if it is
annihilated by the ``discrete Cauchy-Riemann''
{\it partial difference operator} $XY-iY+iX-1$. 
 
Thinking of operators, it was natural to think of them
as {\it polynomials}. While the operator above is
{\it constant coefficients}, one can conceive of linear 
difference operators with
{\it variable coefficients}. A few months later,
also quite by accident (or so it seemed), I leafed through
Rainville's book on Special Functions (reprinted by Chelsea, 1971)
and saw the chapter on Sister Celine's technique. I realized
that with Duffin's operator-notation, things become much
more transparent and generalizable, and that Sister Celine's
technique is essentially a non-commutative analog of
classical elimination methods (notably Sylvester's and
Bezout's) from commutative algebra.
 
Thus was born what later became WZ theory. Herb Wilf added
the crucial notions of {\it WZ pair}, {\it certificate} and {\it duality},
which I would never have been able to conceive
precisely because my random walk was via Dick and Sister
Celine, and Herb's random walk came from elsewhere. \halmos
 
 
\bye
 
 
 
 
 
 
