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 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)):
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.