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Euler prove the Pentagonal number theorem, which is about the denominator of the generating function for the partition function $p(n)$, that is $$\phi(q)=\prod_{k=1}^\infty(1-q^k)=\sum_{n=-\infty}^\infty(-1)^nq^{(3n^2-n)/2}$$

My question is: can we determine the a closed form of $\phi(q)$ for special rational values of $q$? (e.g., $\phi(1/2)$ or $\phi(1/3)$)

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Mathematica give $$\phi(1/2)=(1/2,1/2)_{\infty}=0.2887880950866024$$ where $(a,q)_n$ is the Q-Pochhammer symbol. – user46090 Oct 26 '12 at 14:26
See and for contexts in which that number appears. – joriki Oct 26 '12 at 16:03

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