I found this definition to be different from other ones and the answer makes me feel weird.
If $$A_{N}(x)=\sum_{n=0}^{n=N}a_{n}x^n$$ and $$B_{N}(x)=\sum_{n=0}^{n=N}b_{n}x^n,$$ calculate the coefficients $$c_n$$ in the product $$A_{N}(x)\cdot B_{N}(x)=\sum_{n=0}^{n=N}c_{n}x^n+\sum_{n=N+1}^{n=2N}d_{n}x^n.$$
The answer was $$c_n=a_0b_n+a_1b_{n-1}+…+a_ib_{n-i}+…+a_nb_0.$$
This answer got me pretty lost in thoughts so can someone explain it how did it managed to be this?
Best Answer
It is obtained by distributivity, knowing that, using a more compact notation for the product of polynomials,
$$\biggl(\sum_{n=0}^N a_nx^n\biggr)\biggl(\sum_{n=0}^N b_nx^n\biggr)= \sum_{n=0}^{2N} \biggl(\sum_{\,i+j=n}a_ib_j\biggr)x^n.$$ If $\:0\le n\le N$, for each $i$ such that $0\le i\le n$, $\:j=n-i$ also satisfies the condition $0\le j\le n$, whence the formula $$c_n=\sum_{i=0}^na_ib_{n-i}\qquad\text{if}\quad 0\le n\le N.$$