From: Joerg Arndt <arndt@jjj.de <http://gosper.org/webmail/src/compose.php?send_to=arndt%40jjj.de>> To: math-fun <math-fun@mailman.xmission.com <http://gosper.org/webmail/src/compose.php?send_to=math-fun%40mailman.xmission.com>> Subject: Re: [math-fun] Wilf's pi and determinant problems Message-ID: <20120814160001.GB13398@jjj.de <http://gosper.org/webmail/src/compose.php?send_to=20120814160001.GB13398%40jjj.de>> Content-Type: text/plain; charset=us-ascii * Warren Smith <warren.wds@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=warren.wds%40gmail.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
jj>I guess 1/Pi is somehow verboten, but anyway: 2/Pi = 2F1([-1/2,1/2], [1], 1) 2F1[1/2, 1/2, 3/2, 1] = π/2. More generally, Hypergeometric2F1[1/2, 1/2, 3/2, x^2] = ArcSin[x]/x . wds>--no, no, no: one of Wilf's demands was that the series converge faster than geometrically. wds>For example, Wilf's problem would be trivially solved if he had said not "pi" but "e."
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. The π problem must be very hard, being dangerously close to the algebraic independence of e and π.
Back before complexity theory had a name, Rich determined the time cost of pFq was ≤ "T2" for q≥p, and ≤ "T3" for p=q+1, where T2 was n*cost(n digit multiply) (which we assumed was n log n log log n) and T3 := n*T2. It seemed obvious that π was T3 and e was T2, and proving it would crack algebraic independence. Then Gene's AGM messed everything up by making π T2. But maybe there's still some hay to be made from "hypergeometric independence". --rwg