Well. First of all, I think whatever Gosper is doing about factorials looks pretty brilliant, as fairly usual, but also as usual I do not know how he did it. Second, basically, there was a line of research started by Chowla & Selberg 1967. Here is a recent paper that cites that (I have no idea what the hell this recent paper is doing, but anyway...) http://www.math.wisc.edu/~thyang/Colmez.pdf C&S found some relations between GAMMA(a/b) values and elliptic/modular functions. Putting those together with the usual reflection & n-tupling formulas, then solving the resulting systems of equations, we find out stuff rather like the stuff Gosper is seeking. A number of later authors tried to do exactly that (in 1967 C&S did not have access to Macsyma, etc, handicapping them vs later authors), but you'd have to search the literature to find those (use cite-search on the C&S paper). Now about the transcendence and/or irrationality and/or linear independence (over the algebraic or irrational numbers) results proven by Nesterenko et al, those could sometimes be used to show that there are no further magic relations among the GAMMA(a/b) values... EXCEPT that you'll only be able to show that lack of further relations WITHIN certain classes of allowed kinds of relations. It isn't going to tell you a thing about there being no relations of the form GAMMA(1/5) = exp(sin(GAMMA(3/5)) -- (which is false), but my point is nonlinear nonalgebraic relations cannot be ruled out in this way.
I believe our previous interest in (n/24)! was their rapid computability via Dedekind eta. Even assuming it is always possible to convert to etas, do their modular properties tell all? And, assuming we can canonicalize the eta expression, can we always convert back to Gammas?
--These questions are just new questions... If you were to take the attitude you are not even going to permit elliptic/modular functions to appear, then it might be the reflection and ntupling is all there is and Chowla & Selberg might be irrelevant. -- Warren D. Smith http://RangeVoting.org