[seqfan] Re: Proof or counter sample needed

[Mitch Harris]

On Mon, Oct 27, 2008 at 7:39 AM, Richard Mathar
<mathar at strw.leidenuniv.nl> wrote:
>
>> From seqfan-bounces at list.seqfan.eu  Mon Oct 27 00:50:49 2008
>> Date: Mon, 27 Oct 2008 00:19:31 +0100
>> From: Artur <grafix at csl.pl>
>> To: grafix at csl.pl
>> Cc: seqfan at yahoogroups.com,
>>         Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Subject: Re: [seqfan] Proof or counter sample needed
>>
>> Hypergeometric2F1[1/5, 4/5, 1/2, 3/4]=GoldenRatio=(1+Sqrt[5])/2
>> Artur
>
> This is the case n=3/5, z^2=3/4 of equation 9.121.32 of the
> book by GradStein and Ryshik, namely
> F( (1+n)/2, (1-n)/2 ; 1/2 ; z^2) = [cos(n*arcsin(z))]/sqrt(1-z^2)
> where arcsin[sqrt(3)/2]=Pi/3, see A019863.

OK, this has been bothering me for a week...

How did either of you find figure this out? It's elementary to check,
but -discovering the identities...

- I find hypergeometric series opaque...how did you (Artur) go from
the poly eqn to the hypergeometric, and then how did you get the
golden ratio from that? It looks like magic stated so simply, with
such arbitrary looking parameters.

- How did you (Richard) get the trig solution from Gradstein&Ryzhik?
What pathway led you there?

However these were done, I'm impressed. If by memory/mental
computation, I am not worthy. If by some trick (you just happened to
be skimming G&R and it said it right there), at least I know that's
what it takes.

Mitch