In this paper, it says that the three Borwein cubic theta functions obey the identity $a(q)^{3}=b(q)^{3}+c(q)^{3}$, which is strongly reminiscent of the identity that Dixonian elliptic functions obey $\mathrm{sm}^{3}(z)+\mathrm{cm}^{3}(z)=1$. What relationship (if any) exists between the Dixonian elliptic functions and the Borwein cubic theta functions?
Relationship between Dixonian elliptic functions and Borwein cubic theta functions
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special-functions
analytic-number-theory
theta-functions
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2Hmm, you may be on to something. ${}_2 F_1\left({{\frac13}\atop{}}{{}\atop{\frac43}}{{\frac23}\atop{}}\mid z\right)$ crops up in Dixonian theory, while ${}_2 F_1\left({{\frac13}\atop{}}{{}\atop{1}}{{\frac23}\atop{}}\mid z\right)$ crops up in relating the Borwein theta functions... it might take some tedious algebra to entirely display the connection, though. – 2011-05-03
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1This is a good observation. My student Ng Say Tiong and I have looked at it a few years ago and come up with some results. These are contained in Say Tiong's Master thesis at National University of Singapore. – 2016-09-21
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0Related post. [$f^3+g^3=1$ for two meromorphic functions](http://math.stackexchange.com/questions/29935/f3-g3-1-for-two-meromorphic-functions) – 2017-01-07