If we have a finite sequence a0, a1, . . . , an of real numbers (written briefly as a), then we can look at the polynomial
Pa(t)= a0 + a1t + ··· + antn.
Conversely, given a polynomial Q of degree m ≤ n, we can recover a unique sequence b0, b1, . . . , bn such that
Q (t) = b0 + b1t + · · · + bntn
by, for example, taking bk = Q (k) (0) / k!.
We observe that if a0, a1, . . . ,an , and b0, b1, . . . , bn are finite sequences, then
Pa(t)Pb(t) = pa*b(t),
where a * b = c is a sequence c0, cl, . . . , c2n given by
ck = a0bk + albk-l + . . . + akb0,
where we interpret ai, and bi as 0 if i > n. This sequence is called the convolution of the sequences a and b.
To see the kind of use that one can make of this observation, ...