• Support PF! Buy your school textbooks, materials and every day products Here!

A coefficient problem involving combination

  • Thread starter julypraise
  • Start date
  • #1
110
0

Homework Statement


(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?

Homework Equations


No relevant equation exist.


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.
 

Answers and Replies

  • #2
110
0
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?
 

Related Threads on A coefficient problem involving combination

Replies
2
Views
2K
Replies
5
Views
8K
Replies
3
Views
3K
Replies
2
Views
1K
Replies
4
Views
871
Replies
0
Views
2K
Replies
3
Views
962
Replies
0
Views
4K
  • Last Post
Replies
2
Views
1K
Top