(Combinatorics, mostly:) Given a sequence {an}, one way of expressing the properties of the entire sequence is by defining its generating function

A(z) = a0 + z a1 + z2 a2 + ... + zn an + ...
(For you mathers out there, that's ∑n=0 znan).

A(z) can be defined for real z, or for complex z; it might not be defined for large |z|, and it might not be defined for z != 0... In the last case, however, getting meaningful results out of expressions for A(z) might be tricky. It's still possible when there is some other field or domain where the infinite sum is meaningful.

Now analysing the sequence {an} reduces to analysing A(z). For example, if {an} satisfies a recurrence relation

an = c1 an-1 + c2 an-2 + ... + ck an-k
with ci constant, then A(z) will be a rational function; the converse implication is also true.