Formula for product of two polynomials

Question

I have found such a formula $$ \left(\sum_{k=0}^n a_k z^k \right)\cdot \left(\sum_{j=0}^m b_j z^j \right)=\sum_{k=0}^{n+m} \left( \sum_{j=k-\min(n,m)}^{\min(k,\max(n,m))}a_j b_{k-j} \right)z^k, $$ but seems it's too complicated. Is there any other formulas?

Answer 1

You may simplify a little by omitting the extremes of summation.

Define $a_i=0$ if $i\not\in\{0,1,\dots,n\}$ and $b_j=0$ if $j\not\in\{0,1,\dots,m\}$. Let $c_k$ be the coefficient of $z^k$ in the product. Then the formula is saying that $$ c_k = \sum_{\substack{i,j\in\mathbb N\\i+j=k}} a_ib_j = \sum_{i\in\mathbb N}a_i b_{k-i} . $$ This is as simple as it gets.

This way you can extend the formula to Laurent series.