I was slightly surprised by the following result:

In:= PossibleZeroQ[Gamma[a + b]/Gamma[a] - Pochhammer[a, b]]

Out:= False

versus

In:= PossibleZeroQ[
 Gamma[a + b]/Gamma[a] - Pochhammer[a, b] // FunctionExpand]

Out:= True 

I had the wrong impression (maybe from the name) that PossibleZeroQ would be more likely to err by giving false positives than false negatives. I would think that numerical tests would indeed confirm that these expressions are the same. I guess the problem comes from the poles/zero’s of the Gamma function. Or should I really be careful with FunctionExpand and does it really give possibly false results?

P.S. I came across this in a more practical situation where the equality of two expressions was not easy to see by hand but simplified it to this core oddity.