Multinomial Expansion: Solve (Σ^m_i x(i) Σ^i_j x(i,j))^n

[nikozm]

Hi,

I would like to express the following formula in terms of a straightforward multinomial expansion:

(Σ^{m}_{i=0}x(i)*Σ^{i}_{j=0}x(i,j))^n

Any help would be useful.

 

[WWGD]

Would you explain the notation? What is $x(i)$, are you multiplying the two sums?

 

[nikozm]

Ok, I redifine the expression:

(Σ^{m}_{i=0}x^i*Σ^{i}_{j=0}y^j)^n, where x,y are nonnegative real numbers and m,n are nonnegative integers

 

[mathman]

$(Σ^{m}_{i=0}x^i*Σ^{i}_{j=0}y^j)^n$

It is easier to read as latex. I don't understand what you are looking for. My guess:

y sum is $\frac{1-y^{i+1}}{1-y}$

You can now get i sum to get $(\frac{f(x)-yf(xy)}{1-y})^n\ where\ f(u)=\frac{1-u^{m+1}}{1-u}$