Message: 4 Date: Tue, 14 Aug 2012 02:29:42 -0700 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Wilf's pi and determinant problems Message-ID: <CAA-4O0H75k=nXijRFLy3drYa0yz2BBk=LJ1nr0x-6HpBO8h1fA@mail.gmail.com> Content-Type: text/plain; charset=UTF-8

WDS> Herb Wilf recently died. Here are two open problems he posed.

1. Is there a series for pi=a0+a1+a2+... where a(n) is a rational function of n and a(n-1), which converges faster than geometrically? Equivalently, is there a hypergeometric entire function F(z) with F(1)=pi? --------------- I think you want a0 algebraic. Also, what's special about z=1? But I'm still confused. Suppose a0:=1, a(n):=a(n-1)^2/(n+a(n-1)). What's F? --rwg

Do we even have a (presumably AGMish) recurrence a0 = algebraic, a(n) = algebraicfn(n,a(n-1)), a(?)=??

---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 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. --- Re his #7, for which I solved the "is f(n) of superexponential growth?" part (answer: no, I have exponential lower & upperbounds) I can add the following small addendum: lim f(n)^(1/n) exists and is a number C>1, presumably about 3.85. Proof: because f(n) obeys the property f(a+b) >= f(a) * f(b) by partitioning the nXn matrix into an aXa and bXb block along the (wrong way) diagonal, ignoring the two aXb leftover blocks. Essentially, this stops C from decreasing which in view of my upper bound C<=6.75 forces a limit to exist. This also re-proves my lower bound C>=1338076^(1/13)>2.9598. Q.E.D. I have thought of minor improvements which slightly tighten both my lower & upper bounds, but they are hardly worth describing. A pretty large improvement to the upper bound 6.75 ought to be possible. I have no idea how to find the exact limit value of C. I will write the Univ.Penn. maths dept asking them if they have any idea why Wilf posed his #7.