Difference between revisions of "Generating function for partition function"

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==References==
 
==References==
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Generating function for partition function}}: $24.2.1 \mathrm{I}.A.$
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Partition|next=Closed form for partition function with sinh}}: $24.2.1 \mathrm{I}.B.$
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 20:34, 26 June 2016

Theorem

The following formula holds for $|x|<1$: $$\displaystyle\sum_{k=0}^{\infty} p(k) x^k = \displaystyle\prod_{k=1}^{\infty} \dfrac{1}{1-x^n}=\dfrac{1}{\displaystyle\sum_{k=-\infty}^{\infty}(-1)^k x^{\frac{k(3k+1)}{2}}},$$ where $p(k)$ denotes the partition function.

Proof

References