Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm wondering if there's a special polynomial with a name out there with $x_1,x_2,\ldots,x_k$ as variables that's defined like this:

$$ \sum_{\substack{i_1>0,i_2>0, \ldots,i_k>0 \\ i_1 +i_2+\cdots+i_k=n}} \frac{n!}{i_1! i_2! \cdots i_k!} x_1^{i_1} x_2^{i_2}\cdots x_k^{i_k} $$

share|improve this question
1  
This is the multinomial formula, or multinomial theorem, a generalization of Newton's binomial formula. It is the expansion of $(x_1 + x_2 + ... + x_k)^n$. –  Manolito Pérez Mar 11 '12 at 7:14
2  
@ManolitoPérez I think you guys didn't notice that $i_1>0,i_2>0,...,i_k>0$ –  Tianyang Li Mar 11 '12 at 7:27
1  
@BrianM.Scott I think you guys didn't notice that $i_1>0,i_2>0,...,i_k>0 $ –  Tianyang Li Mar 11 '12 at 7:28
    
My apologies: indeed I did not, despite the title. –  Brian M. Scott Mar 11 '12 at 8:06
2  
This form might help: $$\frac{n!}{(n-k)!}\int_0^{x_1}\cdots\int_0^{x_k} (u_1+\cdots+u_k)^{n-k}du_1\cdots du_k,$$ which appears to also be: $$\sum_{J\subseteq[k]}(-1)^{k-\#J}\left(\sum_{j\in J} x_j\right)^n.$$ –  anon Mar 11 '12 at 8:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.