Call any real that can be evaluated to 2^N decimal places using only N^O(1) arithmetic operations starting from 0 and 1 only, "superfast." Algebraic numbers, pi, e, and formulas you can write with them using exp, log, trig, elliptic and modular functions and their inverse functions, all are superfast. The Gamma function at certain rational arguments, Gamma(a/b), is superfast. For example Gamma(1/2) = sqrt(pi). According to Eric Weisstein, J.Borwein & D.Bailey on p.137 of Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, 2003. show that Gamma(a/24) is superfast for every integer a. I want a better understanding of which rationals and what is going on. Could it be that Gamma(a/b) is superfast for every rational a/b??? I shall end up doubting that. I suspect there are only a FINITE set of such favorable a/b with 0<a<=b. And what is that finite set? ======== integer arguments: Gamma(x) = (x-1)! add-1 formula: Gamma(x+1) = Gamma(x) * x so we need only be interested in Gamma(x) for 0<x<1. reflection formula: Gamma(x) * Gamma(1-x) = pi / sin(pi*x) so we need only be interested in x with 0<x<1/2. k-tuplication & duplication formulae: allow expressing Gamma(k*a/b) in terms of the Gamma(j/b) for integers j with 0<j<=a. Gauss full-period product formula: product(a=1..b)of Gamma(a/b) = b^(-1/2) * (2*pi)^((b-1)/2) proper-fraction subproduct due to Sandor & Toth: product(a=1..b with gcd(a,b)=1)of Gamma(a/b) = (2*pi)^(EulerTotient(b)/2) The product of Gamma(a/b) with a belonging to certain multiplicative subgroups of the integers mod b, can sometimes be expressed in terms of powers of sqrt(pi) and 2, algebraic numbers, and "elliptic singular values" and/or "Dedekind eta function" values. If AGM(1, sqrt(1-k*k)) = sqrt(n) * AGM(1, k) where AGM is the arithogeometric mean and where n is an integer, then k is the "nth singular value" and is superfast. In particular product(a=1..b)of Gamma(a/b)^LegendreSymbol(a|b) can be thus-expressed hence is superfast. See Kabib Muzaffar & KS Williams 2006: http://mathstat.carleton.ca/~williams/papers/pdf/299.pdf James G. Huard, Pierre Kaplan and Kenneth S. Williams: The Chowla–Selberg formula for genera, Acta Arith., 73 (1995) 271-301. JM Borwein & IJ Zucker: Elliptic Integral Evaluation of the Gamma Function at Rational Values of Small Denominator, IMA J. Numerical Analysis 12 (1992) 519-526 A.Selberg & S.Chowla: On Epstein's Zeta-Function, J. reine angew. Math. 227 (1967) 86-110 especially eq2 p.110 which is repeated as eq7 by D.Broadhurst: http://arxiv.org/pdf/0807.2976v3 If b divides 24 or 60 then all Gamma(a/b) values are expressible in terms of only 10 of them at these arguments: 1/3*, 1/4*, 1/5, 2/5, 1/8*, 1/15*, 1/20*, 1/24*, 1/60, 7/60 and for the *-starred ones those Gammas are expressible in terms of elliptic functions: http://arxiv.org/pdf/math/0403510 Let us assume highly optimistically that product(1<=a<=b and a in S)of Gamma(a/b) is superfast whenever S is a multiplicative or additive subgroup of the integers mod b. What would that, plus the k-tuplication and reflection and add-1 formulae, give us? The number of additive subgroups is the same as the number of divisors of b. The number of multiplicative subgroups is upperbounded by the number of divisors of EulerTotient(b/k) summed over all divisors k of b. Now use the known fact that the number of divisors of N is upperbounded by a function that grows more slowly than any positive power of N, and the known fact N/O(logN)<EulerTotient(N)<N. HENCE the total number of such additive or multiplicative subgroups ("number of equations") grows more slowly than any positive power of b. The reflection formula cuts the number of unknowns by a factor of 2 from b to b/2. The k-tuplication formula cuts them by a factor of no more than about log(b), since e.g. Gamma(a/b) if a is relatively prime to b, are immune. The number of unknowns is thus at least about EulerTotient(b)/2. We conclude that even under our optimistic assumption, for every sufficiently large b we are not going to get anywhere near enough equations to determine the unknowns. So in conclusion, I doubt that more than a finite set of a/b allow superfast evaluation of Gamma(a/b). But it might conceivably be possible to do it for all a when b=48, 60, 120, or 240, say [especially favorable numbers with lots of subgroups but small EulerTotient values]. A possible approach to doing so would be to compute product Gamma(a/b) over subgroups to high precision, then attempt to "recognize" the resulting numbers using PSLQ in terms of Dedekind eta function values, algebraic numbers, powers of 2 and pi, and the like. -- Warren D. Smith http://RangeVoting.org