TITLE: product of two polynomials in sigma notation?
QUESTION [7 upvotes]: Say I have polynomials $p(x) = \sum_{k=0}^np_{k}x^k$ and I have $q(x) = \sum_{l=0}^nq_{l}x^l$. I have to write their product in sigma notation. That is $p(x)q(x) = \sum_{k=0}^{2n}A_{k}x^k$ and I have to find $A_{k}$. I have tried writing out the terms for one n = 1 and n = 2. for n = 1 I get:
$p_{0}q_{0} + x(p_{0}q_{1} + p_{1}q_{0}) + x^2p_{1}q_{1}$
and for n = 2 I get:
$p_{0}q_{0} + x(p_{0}q_{1} + p_{1}q_{0}) + x^2(p_{0}q_{2} + p_{1}q_{1} + p_{2}q_{0}) + x^3(p_{1}q_{2} + p_{2}q_{1}) + x^4p_{2}q_{2}$
I've figured out that the sum of the indices of p and q always equal to k. However, I can't seem to put this into sigma notation.

REPLY [4 votes]: $$\sum_{k=0}^{2n}x^k\sum_{j=\max({k-n},0)}^{\min(k,n)}p_jq_{k-j}$$
If you define $p_j=q_j=0$ for $j>n$ you can just write
$$\sum_{k=0}^{2n}x^k\sum_{j=0}^{k}p_jq_{k-j}$$