For the poor, unfortunate souls who have no idea why Gosper said "shew that": http://i.imgur.com/lmFFL.jpg Happy (American-date-convention) pi Day. May your continued fractions forever continue. On Mar 14, 2011, at 9:02 PM, Bill Gosper <billgosper@gmail.com> wrote:
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I saw this sum somewhere as a question as to whether it has a closed form. I am pretty sure it doesn't have any elementary closed form, but I'd think that this isn't a whole new class of functions (with respect to special functions).
(%i1) 'sum(1/(x^k + k!), k, 0, inf); inf ==== \ 1 (%o1) > ------- / k ==== x + k! k = 0
for some real x (we might even just consider positive integral x). Maybe even just
(%i2) 'sum(1/(1 + k!), k, 0, inf); inf ==== \ 1 (%o2) > ------ / k! + 1 ==== k = 0
Can we write this in any other useful forms? (cue the Gosperfunctionology)
-Robert -----------------------
You gotta be kidding. I'm trying to show the kids bilateral infinite matrix products and I can't even "shew that"
Limit[(E^(I N t) (-b + N)! Hypergeometric2F1[1, a - N, b - N, E^(-I t)])/(-a + N)!, N -> \[Infinity]] == -(2^(-a + b) E^(1/2 I (b (-3 Pi + t) + a (Pi + t))) Pi Csc[b Pi]/ ((-Sin[t/2])^(a - b) (-1 + E^(I t)) Gamma[-a + b]))
(integer N, b>a, t>pi/2, and probably more widely).
Where's the nearest home for the bewildered? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun