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A coefficient problem involving combination

  1. Nov 22, 2011 #1
    1. The problem statement, all variables and given/known data
    (1) Is the following formula right?
    [itex]\sum_{l=0}^{m+n} \sum_{k=l-m}^{n} \binom nk \binom {m}{l-k} x^{l} = \sum_{k=0}^{n} \binom nk x^{k} \sum_{j=0}^{m} \binom mj x^{j}[/itex]

    (2) If right, how do I prove it? If not, what is the right formula, and how do I prove it?

    (3) Could you suggest any papers or relevant works that prove this result?

    2. Relevant equations
    No relevant equation exist.


    3. The attempt at a solution
    I've checked that, by specialization of the formula above, the formula is true for some special cases. Actually, this question is originally from Spivak's Calculus, Chapter 2, Problem 4. The problem there, I guess, states a wrong formula, thus I've corrected formula by induction on some special cases.
    I've tried to prove formula by myself, but no progress at all. I've tried a substitution: letting j = l - k on the RHS of the formula, but really no progress. Can there be any lemma, a one really useful to prove this result? Especially, I'm getting trouble on modifying the double sum. I can't change the summand variable in a rigorous way.
     
  2. jcsd
  3. Nov 22, 2011 #2
    After posting, I've got a result:
    [itex](1-x)^{n}(1-x)^{m} = \{ \binom nn x^{0} + \cdots + \binom {n}{l-m} x^{l-m} + \cdots + \binom n0 x^{0} \} \{ \binom m0 x^{0} + \cdots + \binom {m}{l-n} x^{l-n} + \cdots + \binom mm x^{m} \}[/itex]

    If you calculate the coefficient of x^{l} of the above by a product-wise, then you get the answer. Is that right?
     
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