Hi Daniel,

I think this looks okay. The other "hacker" way to do the calculation is:

Normal@Series[ x Exp[   FullSimplify@CoefficientList[
       Normal@  Series[- 2 Pi  /Sqrt[3] Hypergeometric2F1[1/3, 2/3, 1, 1 - x]
           /Hypergeometric2F1[1/3, 2/3, 1, x], {x, 0, 10}], Log[x]][[1]]], {x, 0, 3}]

Outputs:

Out[]:=(1/27)*x + (5 /243)*x^2 + (31/2187)* x^3

And the integerization is as above. There is another question as to whether the output should be a series with rational coefficients or lots of Gamma, PolyGamma whatever. I guess it doesn't matter if FullSimplify then works, nice. You are the expert, so I guess I will just leave it up to you from here, thanks though. This is much better than the original call.

--Brad

Hi Daniel,

Oh no! There was a copy-paste error in the first expansion, it was supposed to be:

Series[Exp[-Pi Hypergeometric2F1[1/2, 1/2, 1, 1 - x]/Hypergeometric2F1[1/2, 1/2, 1, x]], {x, 0, 5}]

and you can test the following by series expansion:

Exp[-Pi Hypergeometric2F1[1/2, 1/2, 1, 1 - x]/Hypergeometric2F1[1/2, 1/2, 1, x]] == EllipticNomeQ[x]

So that first one actually seems to work okay. The other three expansions are more of a concern, they come from Ramanujan's theory, and appear as examples in Bruce Berndt's "Ramanujan's Notebooks Vol. II" ( on p.80-82). Does your fix cover those examples as well?

I may be around the office tomorrow afternoon if you want a better explanation, or to see a few more examples I've cooked up.

--Brad