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Homework Help: Summation with Binomial Expansion

  1. May 9, 2010 #1
    1. The problem statement, all variables and given/known data
    How can i prove this relationship
    \sum _{i=0}^k \text{Binomial}[n+1,k-2i] - \sum _{i=0}^k \text{Binomial}[n,k-2i]=\sum _{i=0}^k \text{Binomial}[n,k-1-2i]



    2. Relevant equations
    Binomial (n,k)=n^k/k!

    3. The attempt at a solution

    I attempted subbing into mathyematica but this didn't work so i attempted by hand and got completely lost. Any helpful comments would be helpful.
    Result from Mathematica
    -Binomial[n, -1 + k] HypergeometricPFQ[{1, 1/2 - k/2,
    1 - k/2}, {1 - k/2 + n/2, 3/2 - k/2 + n/2}, 1] -
    Binomial[n,
    k] HypergeometricPFQ[{1, 1/2 - k/2, -(k/2)}, {1/2 - k/2 + n/2,
    1 - k/2 + n/2}, 1] +
    Binomial[1 + n,
    k] HypergeometricPFQ[{1, 1/2 - k/2, -(k/2)}, {1 - k/2 + n/2,
    3/2 - k/2 + n/2}, 1]
     
    Last edited: May 9, 2010
  2. jcsd
  3. May 9, 2010 #2

    vela

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    Fixed your LaTex. Is this the relation you're supposed to prove?

    [tex]\sum_{i=0}^k \begin{pmatrix}n+1\\k-2i\end{pmatrix} - \sum_{i=0}^k \begin{pmatrix}n\\k-2i\end{pmatrix}=\sum_{i=0}^k \begin{pmatrix}n\\k-1-2i\end{pmatrix}[/tex]

    I don't think it's correct because k-2i<0 for some values of i in the summation.

    That's not right. It should be

    [tex]\begin{pmatrix}n\\k\end{pmatrix}=\frac{n!}{k!(n-k)!}[/tex]
     
  4. May 9, 2010 #3
    but when k-2i<0 the value will be zero
     
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