CoCalc Shared Filesq-polygamma inconsistent on mathematica.sagewsOpen in CoCalc with one click!
Author: Saad Khalid
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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):
Phi2 = lambda z,w: -ln(1-w) + ln(w)*sum((w^(m+z)/(1-w^(m+z))),m,1,10)
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:
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Here it is programmed in Sage:
Phi = lambda x,q: -ln(1-q) + ln(q)*sum((q^(n*x)/(1-q^n)),n,1,10)
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.