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Finding a formula for the multiplication of multiple formal power series

Cbarker1

Active member
Jan 8, 2013
227
MN
Dear Everyone,

I am having trouble with finding a formula of the multiplication 3 formula power series.
\[ \left(\sum_{n=0}^{\infty} a_nx^n \right)\left(\sum_{k=0}^{\infty} b_kx^k \right)\left(\sum_{m=0}^{\infty} c_mx^m \right) \]

Work:

For the constant term:
$a_0b_0c_0$

For The linear term : $(a_1 b_0 c_0 + a_0 b_1 c_0 + a_0 b_0 c_1)x$ + $a_0b_0c_0$

For the quadratic term: $a_2 b_2 c_2 x^6 + a_2 b_2 c_1 x^5 + a_2 b_1 c_2 x^5 + a_1 b_2 c_2 x^5 + a_2 b_2 c_0 x^4 + a_2 b_1 c_1 x^4 + a_1 b_2 c_1 x^4 + a_2 b_0 c_2 x^4 + a_1 b_1 c_2 x^4 + a_0 b_2 c_2 x^4 + a_2 b_1 c_0 x^3 + a_1 b_2 c_0 x^3 + a_2 b_0 c_1 x^3 + a_1 b_1 c_1 x^3 + a_0 b_2 c_1 x^3 + a_1 b_0 c_2 x^3 + a_0 b_1 c_2 x^3 + a_2 b_0 c_0 x^2 + a_1 b_1 c_0 x^2 + a_0 b_2 c_0 x^2 + a_1 b_0 c_1 x^2 + a_0 b_1 c_1 x^2 + a_0 b_0 c_2 x^2 + a_1 b_0 c_0 x + a_0 b_1 c_0 x + a_0 b_0 c_1 x + a_0 b_0 c_0$

I am seeing that the indexes are summing up to the power of x. But how to say that in the indexes?

Thanks,
Cbarker1
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,100
The Astral plane
Say we want to work with the terms [mat]a_2 x^2, ~ b_5 x^5, ~ c_3 x^2[/math]. The product of these terms is \(\displaystyle a_2 b_5 c_3 x^{10}\). If we have a more general term \(\displaystyle a_i x^i b_j x^j c_k x^k = a_i b_j c_k x^{i + j + k}\) you can see the pattern. i + j + k is the power of x involved. Since all we need is the sum we get coefficients \(\displaystyle a_i b_j c_k, ~ a_j b_k c_i, \text{ etc.}\), and we also get any other set of indicies were i + j + k are all the same number, the power of x.

For example, say we want the coefficient of the quartic term of x: \(\displaystyle a_1 b_1 c_2 + a_1 b_2 c_1 + a_2 b_1 c_1\)

Or for the 6th power \(\displaystyle a_1 b_1 c_4 + a_1 b_2 c_3 + a_1 b_3 c_2 + \text{ ...}\).

Does that help? Or am I misinterpreting your question?

-Dan
 

Cbarker1

Active member
Jan 8, 2013
227
MN
Say we want to work with the terms [mat]a_2 x^2, ~ b_5 x^5, ~ c_3 x^2[/math]. The product of these terms is \(\displaystyle a_2 b_5 c_3 x^{10}\). If we have a more general term \(\displaystyle a_i x^i b_j x^j c_k x^k = a_i b_j c_k x^{i + j + k}\) you can see the pattern. i + j + k is the power of x involved. Since all we need is the sum we get coefficients \(\displaystyle a_i b_j c_k, ~ a_j b_k c_i, \text{ etc.}\), and we also get any other set of indicies were i + j + k are all the same number, the power of x.

For example, say we want the coefficient of the quartic term of x: \(\displaystyle a_1 b_1 c_2 + a_1 b_2 c_1 + a_2 b_1 c_1\)

Or for the 6th power \(\displaystyle a_1 b_1 c_4 + a_1 b_2 c_3 + a_1 b_3 c_2 + \text{ ...}\).

Does that help? Or am I misinterpreting your question?

-Dan
I think you misinterpreted my question. For instance, if I have $A$ and $B$ where $A=\sum_{n=0}^{\infty} a_nx^n$ and $B=\sum_{k=0}^{\infty} b_kx^k$, then $$AB=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n} a_nb_{k-n} \right)x^n$$. How can I do the index for the sum with power series?
 

Cbarker1

Active member
Jan 8, 2013
227
MN
From my previous post: How can I do the indices for the sum with 3 power series through the product? I am still confuse by how 3 would work based on the product of 2 power series...
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,648
Leeds, UK
For instance, if I have $A$ and $B$ where $A=\sum_{n=0}^{\infty} a_nx^n$ and $B=\sum_{k=0}^{\infty} b_kx^k$, then $$AB=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n} a_{\color{red}k}b_{\color{red}n-k} \right)x^n.$$ How can I do the index for the sum with power series?
Notice that you have got the indices wrong. For the coefficient of $x^n$ you want the subscripts on $a$ and $b$ to add up to $n$.

Along the same lines, $$ABC=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\left(\sum_{j=0}^{n-k} a_kb_{j}c_{n-k-j} \right)\right)x^n.$$