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