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Statistics summation question:

  1. Oct 26, 2008 #1
    This equation comes out of deriving the canonical partition function for some system. However, the question is more math based. I am having trouble understanding the simplification that was performed in the text:

    ∑ from N=0 to M of: (M!exp((M-2N)a))/(N!(M-N)!) supposedly becomes

    exp(Ma)(1+exp(-2a))^M.... I tried to look at the first few terms and see how I can simplify this, and no dice... Anyone have any ideas? a is just a constant.

    Also, the next step is that the above becomes (2cosh(a))^M Which is great, except, huh? I'm more of a scientist than a math person, so I apologize if I am missing something elementary.

    Thanks!
     
  2. jcsd
  3. Oct 26, 2008 #2

    statdad

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    Homework Helper

    Remember that

    [tex]
    \sum_{n=0}^m \frac{m!}{n!(m-n)!} 1^{(m-n)} b^n = (1 + b)^m
    [/tex]

    In your sum
    [tex]
    \exp\left((M-2N)a\right) = \exp\left(Ma\right) \cdot \left(\exp(-2a)\right)^N
    [/tex]

    so your sum is

    [tex]
    \sum_{N=0}^M {\frac{M!}{N!(M-N)!} \exp\left((M-2N)a\right)}
    = e^{Ma} \sum_{N=0}^M {\frac{M!}{N!(M-N)!} \left(e^{-2a}\right)^N
    = e^{Ma} \left(1 + e^{-2a}\right)^M
    [/tex]

    For the second one - look at the definition of the hyperbolic function as exponentials, and rearrange terms.
     
  4. Oct 26, 2008 #3
    Thanks so much!! That really clears things up.
     
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