Yes:
In[331]:= InverseFourierSequenceTransform[1, x, n, FourierParameters -> {a, 1}] Out[331]= (2 \[Pi])^((1 - a)/2) DiscreteDelta[n]
No:
In[329]:= InverseFourierSequenceTransform[1, x, n, FourierParameters -> {a, 2 Pi}] Out[329]= 0
