Difference between revisions of "Generating function for partition function"
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(Created page with "==Theorem== The following formula holds for $|x|<1$: $$\displaystyle\sum_{k=0}^{\infty} p(k) x^k = \displaystyle\prod_{k=1}^{\infty} \dfrac{1}{1-x^n}=\dfrac{1}{\displaystyle\s...") |
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==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev= | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Partition|next=Closed form for partition function}}: $24.2.1 \mathrm{I}.B.$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Revision as of 00:00, 26 June 2016
Theorem
The following formula holds for $|x|<1$: $$\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
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.2.1 \mathrm{I}.B.$
