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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?

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Hmm, 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. –  J. M. May 3 '11 at 19:41
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