Finding a formula for the multiplication of multiple formal power series

In summary, the conversation discusses finding a formula for the multiplication of three power series by looking at the coefficients of different terms. It is revealed that the indices for the sum must add up to the power of x in order to get the correct coefficient. The conversation also mentions the formula for the product of three power series and how the indices must be arranged in order to get the correct coefficients.
  • #1
cbarker1
Gold Member
MHB
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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
 
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  • #2
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
 
  • #3
topsquark said:
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?
 
  • #4
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...
 
  • #5
Cbarker1 said:
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, $$ABC=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\left(\sum_{j=0}^{n-k} a_kb_{j}c_{n-k-j} \right)\right)x^n.$$
 

FAQ: Finding a formula for the multiplication of multiple formal power series

1. What is the purpose of finding a formula for the multiplication of multiple formal power series?

Finding a formula for the multiplication of multiple formal power series allows us to easily and efficiently perform calculations and manipulations on these series, which are commonly used in fields such as mathematics, physics, and engineering.

2. How do you determine the formula for multiplying multiple formal power series?

The formula for multiplying multiple formal power series is determined by first expanding each series into its individual terms, then multiplying each term of one series by each term of the other series. The resulting terms are then combined and simplified using the rules of exponents and coefficients.

3. Can the formula for multiplying multiple formal power series be applied to any type of series?

No, the formula for multiplying multiple formal power series is specifically designed for formal power series, which are infinite series with a finite number of nonzero terms and a variable as the base.

4. Are there any limitations to the formula for multiplying multiple formal power series?

One limitation of the formula for multiplying multiple formal power series is that it only applies to series with a finite number of terms. Additionally, the series must be convergent in order for the formula to accurately represent the product of the two series.

5. How can the formula for multiplying multiple formal power series be used in real-world applications?

The formula for multiplying multiple formal power series can be used in various real-world applications, such as in signal processing, financial modeling, and data analysis. It allows for efficient calculations and simplifications of complex series, making it a valuable tool in many scientific and mathematical fields.

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