︠4a2e88a9-0e4d-4148-b60e-d221dff93dd7i︠ %html I believe that the formula for q-polygamma given on Mathematica's website may be incorrect.
Here is the formula given by Mathematica:

︡864c2b28-5c9e-47f8-9b73-379cd75549b4︡{"html":"I believe that the formula for q-polygamma given on Mathematica's website may be incorrect.
\nHere is the formula given by Mathematica:\n

\n"}︡ ︠579e5656-5f08-4b59-9237-fdf99a041b61i︠ %html Here is the code used to implement this function on Sage (for reproducability):
︡262ff869-fd76-41a2-8f6f-0fe928843a23︡{"html":"Here is the code used to implement this function on Sage (for reproducability):
\n"}︡ ︠b0fe9d17-9bfc-4418-9494-0b65d5a0a286︠ Phi2 = lambda z,w: -ln(1-w) + ln(w)*sum((w^(m+z)/(1-w^(m+z))),m,1,10) ︡e29e2478-c330-455d-b3b0-1ff7dadbb791︡ ︠25d7ecb9-878a-4496-9162-ed4909d35596i︠ %html Here is a plot of this function on Sage, from z = 1 to 2:

︡06e71d5e-9b9d-4b93-b430-a2e9f977da6e︡{"html":"Here is a plot of this function on Sage, from z = 1 to 2:
\n

\n\n"}︡ ︠ef883945-3cee-4045-ade3-11777a107447i︠ %html Here is the plot that Mathematica claims is it supposed to make(compare from z = 1 to 2)):

︡1b2704e7-3442-4d87-b589-f4972b0f5023︡{"html":"Here is the plot that Mathematica claims is it supposed to make(compare from z = 1 to 2)):\n
\n

\n"}︡ ︠90e127db-af66-41ab-b0b5-4fca4db8babei︠ %html 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:

︡8874e399-ac5f-4661-84b9-b7b60802c281︡{"html":"I found an alternate series function for q-polygamma in this paper:\n
\nhttp://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from95to100.pdf\n
\nIt gives this definition:\n
\n $\"Image$ \n"}︡ ︠f2731686-4c1a-4a92-80d8-08d12b508a1ci︠ %html Here it is programmed in Sage: ︡3dc59ca1-0b9d-44cb-aa58-f2bc52946881︡{"html":"Here it is programmed in Sage:\n\n"}︡ ︠2f0658cf-cba8-486d-8c0a-da4ff68423ee︠ Phi = lambda x,q: -ln(1-q) + ln(q)*sum((q^(n*x)/(1-q^n)),n,1,10) ︡9263b86d-e21f-49b8-8118-f24cb4818877︡ ︠5b0512a3-99e8-4696-8be1-54c17316d140i︠ %html Here is the plot of that definition from z = 1 to 2:

︡c833d7c8-ff4a-4c9b-916e-a8756758152c︡{"html":"Here is the plot of that definition from z = 1 to 2:\n
\n

\n"}︡ ︠cd9ac514-6f09-44fa-8b31-0f83d9945732i︠ %html This fits the curve shows in Mathematica's graph much more closely than the graph of the function in Mathematica's documentation ︡08e47459-70b0-418d-9d17-f9ddaf0bf820︡{"html":"This fits the curve shows in Mathematica's graph much more closely than the graph of the function in Mathematica's documentation\n"}︡ ︠f636a0a3-e0b9-4279-8c3b-2d24c37fec5ci︠ %html 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. ︡c1d1cfef-e657-469e-af9e-12dacc67696b︡{"html":"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.\n"}︡ ︠63af2c48-d004-4b22-9825-9c6a3b13054f︠