- #1
WastedGunner
- 8
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I have a problem that arises in quantum field theory. It involves a problem in combinitorics and about the theory of connected graphs.
Essentially, I am trying to prove an identity involving an exponential of an infinite series with the ways to decompose an integer into the sum of integers.
\exp\left({\sum_{n=1}^\infty a_n}\right) = \sum_{n=0}^\infty \sum_{partitions} \prod_{i=1}^n \frac{a_i^{n_i}}{n_i ! }
Where the partition is over the ways to write n as a sum of integers.
http://en.wikipedia.org/wiki/Partition_(number_theory)
\sum_{i=1}^n in_i = n
If you expand out the first few terms of the right hand side, it looks good.
n=0 (I'm taking it to be 1 just to avoid confusion)
n=1
a_1
n=2
a_2 + \frac{1}{2!}a_i^2
cummulative
a_1 + \frac{1}{2!}a_1^2 + a_2
n=3
a_3 + a_1 a_2 + \frac{1}{3!} a_1^3
cummulative
a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3+ a_2 + a_1 a_2 + a_3
n=4
a_4 + a_1 a_3 +\frac{1}{2!} a_2^2 + \frac{1}{2!} a_1^2 a_2 + \frac{1}{4!} a_1^4
cummulative
a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3 + \frac{1}{4!}a_1^4 + a_2 + \frac{1}{2!} a_2^2 + a_1 a_2 + \frac{1}{2!}a_1^2 a_2 + a_3 + a_1 a_3 + a_4
As you can see, this seems to be systematically giving us the terms of the exponential.
Any thoughts on how to prove this generally?
Essentially, I am trying to prove an identity involving an exponential of an infinite series with the ways to decompose an integer into the sum of integers.
\exp\left({\sum_{n=1}^\infty a_n}\right) = \sum_{n=0}^\infty \sum_{partitions} \prod_{i=1}^n \frac{a_i^{n_i}}{n_i ! }
Where the partition is over the ways to write n as a sum of integers.
http://en.wikipedia.org/wiki/Partition_(number_theory)
\sum_{i=1}^n in_i = n
If you expand out the first few terms of the right hand side, it looks good.
n=0 (I'm taking it to be 1 just to avoid confusion)
n=1
a_1
n=2
a_2 + \frac{1}{2!}a_i^2
cummulative
a_1 + \frac{1}{2!}a_1^2 + a_2
n=3
a_3 + a_1 a_2 + \frac{1}{3!} a_1^3
cummulative
a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3+ a_2 + a_1 a_2 + a_3
n=4
a_4 + a_1 a_3 +\frac{1}{2!} a_2^2 + \frac{1}{2!} a_1^2 a_2 + \frac{1}{4!} a_1^4
cummulative
a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3 + \frac{1}{4!}a_1^4 + a_2 + \frac{1}{2!} a_2^2 + a_1 a_2 + \frac{1}{2!}a_1^2 a_2 + a_3 + a_1 a_3 + a_4
As you can see, this seems to be systematically giving us the terms of the exponential.
Any thoughts on how to prove this generally?
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