Bill Gosper <billgosper@gmail.com> writes:
Mathematica failed to telescope an infinite product and produced a big mess. Plugging in small integers,
Gamma[Root[-17 + 18 #1 - 7 #1^2 + #1^3 &, 1]] Gamma[ Root[-17 + 18 #1 - 7 #1^2 + #1^3 &, 2]] Gamma[ Root[-17 + 18 #1 - 7 #1^2 + #1^3 &, 3]] == 5 Gamma[Root[-5 + 7 #1 - 4 #1^2 + #1^3 &, 1]] Gamma[ Root[-5 + 7 #1 - 4 #1^2 + #1^3 &, 2]] Gamma[ Root[-5 + 7 #1 - 4 #1^2 + #1^3 &, 3]],
Different integers: Gamma[Root[-25 + 21 #1 - 7 #1^2 + #1^3 &, 1]] Gamma[ Root[-25 + 21 #1 - 7 #1^2 + #1^3 &, 2]] Gamma[ Root[-25 + 21 #1 - 7 #1^2 + #1^3 &, 3]] == 10 Gamma[Root[-10 + 10 #1 - 4 #1^2 + #1^3 &, 1]] Gamma[ Root[-10 + 10 #1 - 4 #1^2 + #1^3 &, 2]] Gamma[ Root[-10 + 10 #1 - 4 #1^2 + #1^3 &, 3]]
etc., apparently inexhaustibly. Products and sums of a function over a complete set of algebraic conjugates often come out nice [...]
This is not surprising because in each case the cubics are related by integer translation. Namely: taking x=y+1 in the first cubic x^3 - 7*x^2 + 18*x - 17 yields the second cubic y^3 - 4*y^2 + 7*y - 5 so the Gamma products are related by a factor that's the product of the y's, which is indeed 5. The third and fourth cubics are related (and explained) in the same way. NDE