I would send in this example to Woflram Technical Support (support@wolfram.com) letting them know that you think it's wrong.

They can often determine themselves if it is incorrect or not, but at very least notify the appropiate developer so they can take a look at it and resolve it in the future. 

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.