* Warren Smith <warren.wds@gmail.com> [Aug 14. 2012 21:58]:
From: Joerg Arndt <arndt@jjj.de> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Wilf's pi and determinant problems Message-ID: <20120814160001.GB13398@jjj.de> Content-Type: text/plain; charset=us-ascii
* Warren Smith <warren.wds@gmail.com> [Aug 14. 2012 17:26]:
[...]
---I partially mis-worded the pi problem -- should have said the ratio a(n)/a(n-1) was rational function of n, and a(0)=rational. If none of the pi-experts here can come up with a positive solution, which would be the easy way to solve it, then it is probably a hard problem. One relevant piece of maths is "Siegel E-functions" http://en.wikipedia.org/wiki/E-function
I guess 1/Pi is somehow verboten, but anyway: 2/Pi = 2F1([-1/2,1/2], [1], 1)
--no, no, no: one of Wilf's demands was that the series converge faster than geometrically.
Ouch, that (convergence) got lost in email mangling. So we are looking for a hypergeometric with term ratio p(n)/q(n) such that deg(q) > deg(p). I have never seen such a thing, neither for any logarithmic thing (I consider Pi a logarithm). Those with deg(q) > deg(p) seem to always be (deg(q)=deg(p)+1) exp() (and, obviously, trigs), and (deg(q)=deg(p)+2) Bessel-thingies.
For example, Wilf's problem would be trivially solved if he had said not "pi" but "e."
Via p=1, q=n.
Wilf had a web page of "Herb's open problems" here: http://www.math.upenn.edu/~wilf/website/UnsolvedProblems.pdf and the 2 problems I gave were his #1 and #7.
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