(adsbygoogle = window.adsbygoogle || []).push({}); 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A coefficient problem involving combination

**Physics Forums | Science Articles, Homework Help, Discussion**