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I had hoped that this would be easier.

I have power series that I'm using. I want to know what simple structure describes the operations that I'm performing on them. It would be better if I could find something that non-mathematicians (actually computer scientists) would understand, but I'd like to know the technical terminology too.

First, let me define three power series, $f(x)$, $g(x)$, and $h(x)$:

$f(x) = \sum_{i=m_1}^{n_1}{c_i x^i}$

$g(x) = \sum_{i=m_1}^{n_1}{d_i x^i}$

$h(y) = \sum_{i=m_2}^{n_2}{e_i y^i}$

Then the two operations that I perform are as follows:

(1) Subtraction $r(x) = f(x) - g(x)$

$r(x) = \sum_{i=m_1}^{n_1}{c_i - d_i}$

(2) Multiplication $r(x,y) = f(x)h(y)$

$r(x,y) = \sum_{i=m_1}^{n_1}{\sum_{j=m_2}^{n_2}{c_i \cdot e_j}}$

All of the coefficients and variables are integers.

I'm writing a paper, and I'm searching for a more "definite" description than what I have. It is somewhat ackward to describe, and I want to convey this information exactly and concisely. Plus, I don't want the reader to have any doubts.

I've been considering that I should just write out the math, since it seems to leave less doubt.

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    In your description, r(x) and r(x,y) do not depend on x, but only on the coefficients of f,g,h. Is it a typo or do you really mean that r(x) and r(x,y) are constants and not power series?2011-05-19

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The first is $\rm\ (f-g)(1)\ =\ f(1)-g(1)\ $ and the second is $\rm\ (f\:h)(1)\ =\ f(1)\ h(1)$

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    Here's how I get series. Take the integers from 0 to 9, for example. I look at the binary digits. The least significant bit of these numbers is 0,1,0,1,0,1,0,1,0,1. This is one series. The next least significant bit is 0,0,1,1,0,0,1,1,0,0. This is another series. The trick is that I use the product repeatedly to get an $n$-dimensional construct (or multivariate series). I want to call this a matrix or grid, but I doubt that this conveys the operations well. Again, an analogy would be excellent.2010-11-08