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.