* rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Aug 28. 2008 13:29]:
In case anybody besides me missed it,
Gamma(1/3) = 2^(11/18)*(sqrt(3)+1)^(1/3)*%pi^(2/3)/(3^(1/4)*agm(sqrt(2)*(sqrt(3)+1),1)^(1/3))
11/18 1/3 2/3 1 2 (sqrt(3) + 1) %pi gamma(-) = ------------------------------------- 3 1/4 1/3 3 AGM (sqrt(2) (sqrt(3) + 1), 1)
follows from Borwein&Borwein, pp 15(a) and 28(d), so Gamma(1/3) is nearly as easy as pi. Similarly,
Gamma(1/4) = 2^(3/4)*%pi^(3/4)/sqrt(agm(sqrt(2),1))
3/4 3/4 1 2 %pi Gamma(-) = --------------------- 4 sqrt(AGM(sqrt(2), 1))
finishes off Gamma(n/12). Interesting that, unlike pi, there are no hypergeometric series for these, although the sum(n/(%e^(2*sqrt(3)*%pi*n)-1),n,1,inf) is a degenerate bibasic series.
Joerg Arndt> cf. http://arxiv.org/abs/math/0403510 (section 5)
gives similar expressions also for gamma(1/8) and gamma(1/24)
That's the sort of thing I'd much prefer be done by an active CAS. Same goes for most of http://functions.wolfram.com . But it's probably too hard to automate the singular value stuff, so we're stuck with tables. (Of course, we could hide tables inside the CAS. Macsyma once had an error: "Factor ran out of primes!") Likewise for the singular eta' values I'm finding, and their consequent eta'/eta sums: 'Sum(n/(%e^(2*sqrt(7)*%pi*n)-1),n,1,inf) = -9*Gamma(1/7)^2*Gamma(2/7)^2*Gamma(4/7)^2/(3584*%pi^4)-1/(8*sqrt(7)*%pi)+1/24 inf 2 1 2 2 2 4 ==== 9 Gamma (-) Gamma (-) Gamma (-) \ n 7 7 7 > --------------------- = - ------------------------------- / 2 sqrt(7) %pi n 4 ==== %e - 1 3584 %pi n = 1 1 1 - ------------- + -- 8 sqrt(7) %pi 24 'Sum(n/(%e^(2*%pi*n/sqrt(7))-1),n,1,inf) = 9*Gamma(1/7)^2*Gamma(2/7)^2*Gamma(4/7)^2/(512*%pi^4)-sqrt(7)/(8*%pi)+1/24 inf 2 1 2 2 2 4 ==== 9 Gamma (-) Gamma (-) Gamma (-) \ n 7 7 7 sqrt(7) 1 > ------------- = ------------------------------- - ------- + -- / 2 %pi n 4 8 %pi 24 ==== ------- 512 %pi n = 1 sqrt(7) %e - 1 Unfortunately, tables have bugs. E.g., for K(k_11) in terms of Betas, Eric defines a constant R as "the real root of x^3-4x=4=0". The obvious guesses x^3-4x=4 and x^3-4*x+4=0 don't seem to work. Earlier, they use the undenested radicals sqrt(2+-sqrt(3)), which also make me nervous. --rwg rwg>Does anyone besides Salamin remember the meaning of the agm error in
Peter Samson's (1960s) PDP-1 music compiler?
TK> That argument greedily masticates. Would you believe Samson himself had forgotten this? Seemed incredulous, even. It suggests that they don't have that compiler at the restoration project.
[... AGM and eta ...]
lovely!
Tnx!