From o.warnaar@maths.uq.edu.au Thu Jan 3 21:20:13 2013
Dear Doron,
Thanks very much for your very kind words about my little survey. I am glad
you though it (mostly) beautiful.
I do feel I have to defend (hopefully without having to attack you) the
great analysts Andrews, Askey, Hardy and Ismail (as well as my own writing
of course).
First of all, when I was asked to write a short survey on the 1psi1 sum in
honour of Ramanujan (and Andrews-Askey-Hardy-Ismail-etc) I did not think
the editors had in mind giving me the opportunity to get on my hobby-horse
and lecture about the pros and cons of the analytic versus the formal
point of view of hypergeometric series. Rather what I tried to do is to
give an reasonably accessible and complete account of the current state of
the art of the 1psi1 sum.
Of course you are right that for proving many a result in combinatorics
it is perfectly ok to take a purely formal point of view.
However, if you want to think of, say, the q-binomial theorem
\sum_k (a)_k z^k/(q)_k = (az)_{\infty}/(z)_{\infty}
as a formal power series in z (perhaps in order to prove your favourite
combinatorial identity) then no harm is done if you first learned about
this identity from one of the above-mentioned great analysts as an analytic
identity true for |z|<1. You can still do your formal manipulations of
course, you simply ignore the fact that Dick or George or Mourad wrote this
annoying |z|<1 condition.
Conversely, I do not think your brilliant new proof of
1+1+\dots+1=n+1 (*)
is similar to what you consider a flawed proof of Jacobi's 4 squares
theorem using the 1psi1 sum. Note that the proof that I presented is not
only shorter than your (nonetheless brilliant) proof of (*) but I challenge
you to find a shorter proof NOT using the 1psi1 summation. To make this a
real challenge, as I argued above, I do not consider a proof that is
identical to the one I gave but which drops the |b/a|<|z|<1 condition in
the statement of the 1psi1 sum as being different (though I admit it is
shorter by 11 characters). My challenge on the other hand will be to find a
shorter proof of (*), so the gloves are off!
Finally, to the formal power series point of view, because that's what it
really is about. I just read your "fix" of Mourad's proof from the book.
Of course, by definition, proofs from the book don't need fixing, so I
think Mourad is safe.
What you are doing is of course less general because there are examples of
formal identities for which your (again brilliant) fix-of-Mourad's-proof
method works but which do NOT correspond to a meaningful analytic identity,
and, regrettably, in some rare case (as George pointed out in his book)
analytic identities are important (to some of us).
Finally, I sincerely apologise if my survey gave the impression that in
order to prove combinatorial claims like those of Jacobi requires analytic
identities. I am quite confident however that most analysts can see through
my flawed presentation. The analysts I know are all quite clever, almost as
clever as the formalists I know.
Hartelijke groet & a happy 2013,
Ole
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