Difference between revisions of "Generating function for partition function"
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Revision as of 23:59, 25 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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.2.1 \mathrm{I}.A.$
