Put x = 1/18 and x = 17/90 in the identity:
gamma(x)*gamma(x+1/3)*gamma(x+2/3) = 2*pi*3^(1/2-3x)*gamma(3x),
and multiply the results together.
Warut
[Also caught by Dick Askey]
[And Rich, whose message was delayed somehow.]
Gack! Macsyma's and Mathematica's Gamma simplifiers are supposed to try that, so there are bugs in both! --rwg I was lazy to trust them.
Rich pointed out http://arxiv.org/abs/0907.4384, which gives a simple formula for Product Gamma(k/n) 1<=k<=n (k,n)=1 This seems to generalize to /===\ /===\ | | k sqrt(2 pi) phi(n) | | mu(d) n z | | Gamma(z + -) = (-----------------) | | Gamma (---), | | n 1 | | d 1<=k<=n /===\ ----- d | s(n) (k,n)=1 | | p - 1 z (n | | p ) | | p prime p | n where s(n) is the squarefree part of n, phi is Euler's totient, and mu is Moebius's trivalue. For (sort of) legible, see http:\\gosper.org\gamprd.html. (Vertical derangement not in .nb). Just two Gammas for n a prime power, p - 1 ----- 2 m - 1 (2 pi) p m sqrt(p) (------------------) Gamma(p z) /===\ (m (p - 1) + 1) z | | k p | | Gamma(z + --) = ----------------------------------------------, | | m m - 1 m p Gamma(p z) 1<=k<=p (k,p)=1 and for powers of 2, just one: /===\ n - 2 n - 1 | | k 2 - 1/2 2 (1 - n) z n - 1 1 | | gamma(z + --) = (2 pi) 2 gamma(2 z + -) | | n 2 n 2 1<=k<=2 k odd The paper also gives prod Gamma(Farey sequence), which presumably generalizes as above. And both identities should have prod|sum trig analogs, e.g. sum(tan(x+k/n)), (k,n)=1. --rwg