[seqfan] Re: Proof or counter sample needed

[Richard Mathar]

mh> From seqfan-bounces at list.seqfan.eu  Wed Nov  5 17:00:58 2008
mh> Date: Wed, 5 Nov 2008 10:45:26 -0500
mh> From: "Mitch Harris" <maharri at gmail.com>
mh> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
mh> Subject: [seqfan] Re: Proof or counter sample needed
mh> 
mh> On Mon, Oct 27, 2008 at 7:39 AM, Richard Mathar
mh> <mathar at strw.leidenuniv.nl> wrote:
mh> >
mh> >aj> From seqfan-bounces at list.seqfan.eu  Mon Oct 27 00:50:49 2008
mh> >aj> Date: Mon, 27 Oct 2008 00:19:31 +0100
mh> >aj> From: Artur <grafix at csl.pl>
mh> >aj> To: grafix at csl.pl
mh> >aj> Cc: seqfan at yahoogroups.com,
mh> >aj>         Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
mh> >aj> Subject: Re: [seqfan] Proof or counter sample needed
mh> >aj>
mh> >aj> Hypergeometric2F1[1/5, 4/5, 1/2, 3/4]=GoldenRatio=(1+Sqrt[5])/2
mh> >aj> Artur
mh> >
mh> > This is the case n=3/5, z^2=3/4 of equation 9.121.32 of the
mh> > book by GradStein and Ryshik, namely
mh> > F( (1+n)/2, (1-n)/2 ; 1/2 ; z^2) = [cos(n*arcsin(z))]/sqrt(1-z^2)
mh> > where arcsin[sqrt(3)/2]=Pi/3, see A019863.
mh> 
mh> ....
mh> 
mh> How did either of you find figure this out? It's elementary to check,
mh> but -discovering the identities...
mh> 
mh> - I find hypergeometric series opaque...how did you (Artur) go from
mh> the poly eqn to the hypergeometric, and then how did you get the
mh> golden ratio from that? It looks like magic stated so simply, with
mh> such arbitrary looking parameters.
mh> 
mh> - How did you (Richard) get the trig solution from Gradstein&Ryzhik?
mh> What pathway led you there?
mh> 
mh> However these were done, I'm impressed. If by memory/mental
mh> computation, I am not worthy. If by some trick (you just happened to
mh> be skimming G&R and it said it right there), at least I know that's
mh> what it takes.
mh> ....

This just expresses that I've been confronted
with the hypergeometric functions lately. You'll find them in full-fledged
use in my preprints in

http://arxiv.org/abs/0809.2368 (first appearance on page 1, then 4F3 page 11,
                       next page 13)
http://arxiv.org/abs/0805.3979 (page 6)
http://arxiv.org/abs/0705.1329 (page 3, 4, 5)

Grabbing the Gradstein-Ryzhik book has become a form of reflex to me;
I've also made it a habit to accumulate them in my local list of
mathematical formulas, see sections 9.121 and 9.14 in
http://www.strw.leidenuniv.nl/~mathar/public/mathar20071105.pdf

Let me stress that all the orthogonal polynomials which we (=physicists)
meet every day in applications (quantum mechanics, heat equations,
acoustics, laser beam waists) are a subset of these Gaussian hypergeometric
functions. From that "application" point of view they are much more frequent
than pure mathematicians would guess. Solving some 2nd order DEQ with
a series ansatz (= the desperate resort) often ends up with that sort
of hypergeometric solution in a wide class of linear, important cases.

To the benefit of other people who might meet that specific formula,
I've added comments to the OEIS sequences with these sines/cosines that
may be found by using "Mathar 2F1" in the OEIS search window.

But aside from that, basically, this type of result is pure luck.
-- 
Richard J Mathar                Tel (+31) (0) 71 527 8459
Sterrewacht Universiteit Leiden Fax (+31) (0) 71 527 5819
Postbus 9513
2300RA Leiden
The Netherlands                 URL http://www.strw.leidenuniv.nl/~mathar
office: Niels Bohrweg 2, 2333 CA Leiden, #101