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. E.g., Out[654]= (E^(-((2 I Pi)/3)) z)! (E^((2 I Pi)/3) z)! == (Pi Csc[E^(-((I Pi)/3)) Pi z])/Pochhammer[1 + E^(-((2 I Pi)/3)) z, -1 + z] In[655]:= Table[ N[Divide @@ %], {z, {-1, 1/3, 1/2, 1, 6, 9, E, Pi, I}}] Out[655]= {0.9999999999999925 - 9.025031411218348*10^-18 I, 1.000000000000002 - 2.272641409138545*10^-16 I, 1. + 0. I, 0.9999999999999911 + 3.885780586188048*10^-16 I, 0.9999999999999996 + 4.33801051959268*10^-15 I, 0.9999999999999991 + 4.773959005888173*10^-15 I, 0.9999999999999993 + 8.129524651428513*10^-16 I, 0.9999999999999978 + 6.339440984328298*10^-16 I, 1.000000000000002 + 2.775557561562891*10^-17 I} Similarly for Gammas of cubic surds? --rwg