This appears to be a faulty defnition in mathematica, I'm using mathematica 8.
Using the Following code:
Product[(m + i) (m - (i + 1)/2)/2, {i, 0, m - 2}]
FullSimplify[%]/%
% /. m -> 3
I get the answers:
(-1)^m 4^(
1 - m) m (-1 + 2 m) Pochhammer[2 - 2 m, -2 + m] Pochhammer[
1 + m, -2 + m]
(4^m Sqrt[\[Pi]]
Gamma[-1 + 2 m])/(m (-1 + 2 m) Gamma[1/2 - m] Gamma[
1 + m] Pochhammer[2 - 2 m, -2 + m] Pochhammer[1 + m, -2 + m])
2
The original form is correct, the simplified version contains an incorrect factor of 2. If non-integers are used in the replace the ratio becomes equal to 1 only because both sides are then incorrect by the same factor (Should be 15, not 30). Using "Assume m is an integer... " does not simplify to Gamma functions and therefore doesn't have the same problem.
This feels like a faulty defintion in mathematicas simplify commands, or have I missed something?