I used Table[] to see the values of your function:

In[64]:= 
Table[pn2[n] /. {\[Alpha] -> 1/2, \[Beta] -> 1/2 }, {n, 0, 10}]

Out[64]= {{0.5, 0.5, 6.93889*10^-18}, {0.4375, 0.5, 0.0625}, {0.40625,
   0.53125, 0.0625}, {0.402344, 0.53125, 0.0664063}, {0.400391, 
  0.533203, 0.0664063}, {0.400146, 0.533203, 0.0666504}, {0.400024, 
  0.533325, 0.0666504}, {0.400009, 0.533325, 0.0666656}, {0.400002, 
  0.533333, 0.0666656}, {0.400001, 0.533333, 0.0666666}, {0.4, 
  0.533333, 0.0666666}}

There are no complex values, but you must use ParametricPlot3D[]. Unfortunately, it doesn't plot the graph. At least, ListPlot3D[] gives some appropriate image.

Ok you see I did a list of pn2 values for naturals from 0 to 10 and get real vectors. It seems the only complex values occur when n is not an integer. I'm now trying to use discrete plot....thank you very much, I didn't notice plot is continuous, which is pretty obvious. Thanks!