# [seqfan] Re: Proof or counter sample needed

Richard Mathar mathar at strw.leidenuniv.nl
Fri Nov 7 22:03:24 CET 2008

```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,
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

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