Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\phi(1/3)$)

share|cite|improve this question
    
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
1  
See math.stackexchange.com/a/54248 and math.stackexchange.com/a/219978 for contexts in which that number appears. – joriki Oct 26 '12 at 16:03

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.