On Thu, Feb 14, 2013 at 12:48 PM, Bill Gosper <billgosper@gmail.com> wrote:
Decades ago in an MIT talk, I gave a method to enumerate a family of telescoping products akin to Product[(1/2)*(1 + x^2^(-n)), {n, Infinity}] == (-1 + x)/Log[x] . (Which telescopes because you can shift the index by changing the x variable.)
Recently, Mourad Ismail sent a paper reminding me to carry out the enumeration, on which Neil and I started yesterday. Neil found
Product[(1/2)*(z^(-2)^(-k) + 1), {k, 1, Infinity}] == (z - 1)/(z^(2/3)*Log[z])
and fears it to be centuries old, after getting severely scooped on a continued fraction identity last week.
I found
Product[-(-1)^(2/3) + (-1)^(1/3)*z^(-2)^(-k), {k, Infinity}] == (-1 + (-1)^(2/3)*z)/((-1 + (-1)^(2/3))*z^(2/3))
which looks oscillatory and nonconvergent, but is really just equivalent to
Product[2*Sin[Pi/6 + (-1/2)^k*t], {k, Infinity}]==(2*Cos[Pi/6 + t])/Sqrt[3]
from http://www.tweedledum.com/rwg/idents.htm . (Nice(?) exercise.)
Two of the above identities come from the scheme (a*x^4 + b*x^2 + c)/(c*x^2 + b*x + a) reducing to a polynomial, which Reduce promises contains three more. And there are other schemes... --rwg
The antepenultimate scheme is (1+x^2+x^4)/(1+x+x^2) == 1-x+x^2 . But this is just (1+x^3)/(1+x), so we can divide Neil's result into itself with z cubed: Product[1 - z^(-2)^(-k) + z^(2^(1 - k)/(-1)^k), {k, 1, Infinity}] == (1 + z + z^2)/(3*z^(4/3)) and this is equivalent to Product[-1 + 2*Cos[t/(-2)^k], {k, 1, Infinity}] == (1/3)*(1 + 2*Cos[t]) which is already in idents.htm . Only two more shots at novelty. In this scheme. --rwg bob baillie>in any case, video like this:http://www.guardian.co.uk/science/2013/feb/15/meteorite-explodes-over-russia... is pretty amazing! And just what I was looking for. http://www.youtube.com/watch?feature=player_embedded&v=gQ6Pa5Pv_io#! Flash: 4:37; Bang: 7:00 But can someone explain the multiplicity of bangs that characterize these bolides? Is it just sonic booms from individual fragments on different trajectories, or maybe similar trajectories with different speeds? And what's with the bifid symmetry? Maybe this is all just a promotion from the Beijing Cruller and Fireworks Collective. Or is it a natural, convective consequence of heating a horizontally cylindrical region of a fluid? And speaking of heat, I haven't heard anyone report feeling the heat of the flash. "Finally", in this heavily viewed video, http://www.youtube.com/watch?v=d5xMYRBpLSI , the driver hangs a right and seems to chase the meteor at high speed. But notice where the cloud disappears behind his(?) roof edge. It's drifting way faster in that same direction. I thought there was little wind in the stratosphere. Did the meteor impart a bunch of momentum? Energy must be fairly cheap in Chelyabinsk to justify so many single pane windows in such a frigid climate. A harbinger of prosperity.