Difference between revisions of "Pure recurrence relation for partition function"
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(Created page with "==Theorem== The following formula holds: $$p(n)=\displaystyle\sum_{1 \leq \frac{3k^3 \pm k}{2} \leq n} (-1)^{k-1} p \left( n - \dfrac{3k^2 \pm k}{2} \right) = \dfrac{1}{n} \di...") |
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$p(n)=\displaystyle\sum_{1 \leq \frac{3k^3 \pm k}{2} \leq n} (-1)^{k-1} p \left( n - \dfrac{3k^2 \pm k}{2} \right | + | $$p(n)=\displaystyle\sum_{1 \leq \frac{3k^3 \pm k}{2} \leq n} (-1)^{k-1} p \left( n - \dfrac{3k^2 \pm k}{2} \right),$$ |
where $p(n)$ denotes the [[partition]] function and $\sigma_1$ denotes the [[sum of divisors]] function. | where $p(n)$ denotes the [[partition]] function and $\sigma_1$ denotes the [[sum of divisors]] function. | ||
| Line 7: | Line 7: | ||
==References== | ==References== | ||
| − | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Closed form for partition function|next= | + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Closed form for partition function with sinh|next=Recurrence relation for partition function with sum of divisors}}: $24.2.1 \mathrm{II}.A.$ |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 20:41, 26 June 2016
Theorem
The following formula holds: $$p(n)=\displaystyle\sum_{1 \leq \frac{3k^3 \pm k}{2} \leq n} (-1)^{k-1} p \left( n - \dfrac{3k^2 \pm k}{2} \right),$$ where $p(n)$ denotes the partition function and $\sigma_1$ denotes the sum of divisors function.
Proof
References
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $24.2.1 \mathrm{II}.A.$
