That was my first thought too, in which case there should be a warning that the solution is only valid for non integers.

The reason I dont think this is the case is two fold:

1.) The simplified function with Gamma's is continuous throughout all real values I've tried and is exactly twice the Pochhammer version at all integers tested. 30 for m=3.

2.) The Pochhammer function is primerily defined as a function on the integers, which gives a value of 15 above. Trying to input a non integer into the pochhammer version converts it to Gamma's automatically. This results in a function tending to the 30 in the limit of m=3, but having the value of 15 at m=3. This therefore gives a discontinuous function which exactly halfs its value at the integers. The only values I can test are at the integers which I'm pretty sure are correct for the Pochhammer version, and although I don't know how the function should behave, this seems particularly strange behaviour.

It just feels like whever wrote up the conversion missed a factor of two in their derivation. Interestingly this only works for this very specific equation, any variation I tried resulted in either the correct answer or indeterminate value.