I believe that the formula for q-polygamma given on Mathematica's website may be incorrect.

Here is the formula given by Mathematica:

Here is the formula given by Mathematica:

Here is the code used to implement this function on Sage (for reproducability):

Here is a plot of this function on Sage, from z = 1 to 2:

Here is the plot that Mathematica claims is it supposed to make(compare from z = 1 to 2)):

I found an alternate series function for q-polygamma in this paper:

http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from95to100.pdf

It gives this definition:

http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from95to100.pdf

It gives this definition:

Here it is programmed in Sage:

Here is the plot of that definition from z = 1 to 2:

This fits the curve shows in Mathematica's graph much more closely than the graph of the function in Mathematica's documentation

Please tell me if you see any errors. However, I believe that Mathematica may have made an error in its documentation of the Q-Polygamma function. Thank you for your time.