1

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 $\phi(1/3)$)

sperners lemma
  • 2,259
AmFCG
  • 581
  • 2
  • 12
  • Mathematica give $$\phi(1/2)=(1/2,1/2)_{\infty}=0.2887880950866024$$ where $(a,q)_n$ is the Q-Pochhammer symbol. – AmFCG Oct 26 '12 at 14:26
  • 1
    See http://math.stackexchange.com/a/54248 and http://math.stackexchange.com/a/219978 for contexts in which that number appears. – joriki Oct 26 '12 at 16:03

0 Answers0