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. ((1 = AGM(eta(q^2)^10/(eta(q)^4*eta(q^4)^4),eta(q)^4/eta(q^2)^2)) = AGM((16*eta(q^4)^8+eta(q)^8)^(5/12)/eta(q)^(2/3),eta(q)^(10/3)/(16*eta(q^4)^8+eta(q)^8)^(1/12))/eta(q^4)^(2/3)) = AGM(sqrt(64*eta(q^4)^24+eta(q^2)^24)-8*eta(q^4)^12,eta(q^2)^12)/(eta(q^2)^2*eta(q^4)^2*sqrt(sqrt(64*eta(q^4)^24+eta(q^2)^24)-8*eta(q^4)^12)) 10 2 4 eta (q ) eta (q) 1 = AGM(----------------, --------) 4 4 4 2 2 eta (q) eta (q ) eta (q ) 8 4 8 5/12 10/3 (16 eta (q ) + eta (q)) eta (q) AGM(---------------------------, ---------------------------) 2/3 8 4 8 1/12 eta (q) (16 eta (q ) + eta (q)) = ------------------------------------------------------------- 2/3 4 eta (q ) 24 4 24 2 12 4 12 2 AGM(sqrt(64 eta (q ) + eta (q )) - 8 eta (q ), eta (q )) = --------------------------------------------------------------------. 2 2 2 4 24 4 24 2 12 4 eta (q ) eta (q ) sqrt(sqrt(64 eta (q ) + eta (q )) - 8 eta (q )) --rwg Does anyone besides Salamin remember the meaning of the agm error in Peter Samson's (1960s) PDP-1 music compiler?