What structure is this?

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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I'm still considering that I could just call this subtraction and products of power series...But it seems to leave things open to the imagination:-( – Matt Groff Nov 8 '10 at 2:27
It's still not clear what you mean. Why don't you give some specific examples. – Bill Dubuque Nov 8 '10 at 3:00
I guess I may be able to refer to the functions or series as $n$-dimensional hyperplanes. For series with one variable, this is a one dimensional hyperplane which corresponds to a line. The products transform an $m$ and $n$ dimensional hyperplane into an $n+m$ dimensional hyperplane. It seems complex, though. – Matt Groff Nov 8 '10 at 8:03
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? – ogerard May 19 '11 at 8:33

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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Do these have names? I'm still hoping to get a good verbal description... – Matt Groff Nov 8 '10 at 3:41
@Matt: They might in some contexts. What is yours? E.g. f(1) is the digit sum if f is a radix polynomial. – Bill Dubuque Nov 8 '10 at 3:53
I'm using something like a generating function for a computer bit in a series of numbers. For example, take a series of numbers from zero to some natural $u$. Now look at the bits that represent them. I pick the $v$th least significant bit, and represent it. – Matt Groff Nov 8 '10 at 4:44
@Matt: Again, a concrete example would help to clarify your intent. – Bill Dubuque Nov 8 '10 at 4:57
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. – Matt Groff Nov 8 '10 at 5:55