Multinomial Expansion in Mathematica

[EngWiPy]

Hello,

I have the following equation, which is simply the multinomial exapnsion:

$$\left[\sum_{l=1}^L\,x_l\right]^n=\sum_{k_1,k_2,\ldots,k_L}{n\choose k_1,k_2,\ldots,k_L}\prod_{l=1}^L\,x_l^{k_l}$$

where the summation is taken over all possibilities that the summation of the nonnegative integer indices k1,k2,...,kL equal n.

How can I program this using Mathematica??

Thanks in advance

 

[EngWiPy]

Any suggestion please? I am really need this. Any help will be greatly appreciated.

 

[Hurkyl]

It really isn't clear what you want to do.

 

[EngWiPy]

It really isn't clear what you want to do.

Ok, first of all, thank you for replying. What I want to do is to write a function using Mathematica that computes the multinomial expansion, which is the right hand side equation in the first post. The difficulty as I see it, comes from the summation. For example:

$$(x_1+x_2+x_3)^2=\underbrace{\sum_{k_1,k_2,k_3}{2\choose k_1,k_2,k_3}\prod_{n=1}^{3}x_n^{k_n}}_{\text{Multinomial Expansion}}$$

where the summation in the right hand side takes place over all possibilities that $$k_1+k_2+k_3=2$$, and
$$\left(\begin{smallmatrix}2\\k_1,k_2,k_2\end{smallmatrix}\right)=\frac{2!}{k_1!\,k_2!\,k_3!}$$

Doing that yields to the final result:

$$(x_1+x_2+x_3)^2=x_1^2+x_2^2+x_3^2+2x_1x_2+2x_1x_3+2x_2x_3$$


I hope that I explain the point.

Regards

 

[Hurkyl]

What's wrong with the Expand function?

 

[EngWiPy]

What's wrong with the Expand function?

The problem is that, the multinomial I have looks like this:

$$\left[\sum_{n=1}^NA_n\,\tex{e}^{-\gamma/\gamma_n}\right]^m$$

I want to exctract the exponentials and combine them with other exponentials and integrate them all over $$\gamma$$ to do my derivation for the performance metrics.

 

[EngWiPy]

So far I reached to this point:

x = 0;
For[k1 = 0, k1 <= 2, k1++,
 For[k2 = 0, k2 <= 2, k2++,
  For[k3 = 0, k3 <= 2, k3++,
   If[k1 + k2 + k3 == 2,
    x = x + Multinomial[k1, k2, k3]*x1^k1*x2^k2*x3^k3];
   If[k1 == 2 && k2 == 2 && k3 == 2, Print[x]]]]]

x1^2+2 x1 x2+x2^2+2 x1 x3+2 x2 x3+x3^2

But how can I write it in general form, when the number of terms and the exponent are variables?

 

[Hurkyl]

I'm still not sure why you think this will result in anything different than the expand function...


When the number of loops is variable, you essentially have to implement an odometer -- e.g. as a list whose n-th element is the value of the n-th variable. Then you have just one for loop that initializes / steps this odometer.


It might be easier, however, to construct the m-fold Cartesian product of the set {0, 1, ..., N} with itself. There is surely a builtin function to compute Cartesian products of lists, and it should be easy enough to use that to write a function that iterates it m times. But I don't have mathematica on my home computer to tell you what that function would be.

(The Cartesian product of A with B is the set AxB whose elements are all length-2 lists whose first element is from A and whose second element is from B)