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Let us say we are given two polynomials $p=\sum_{k=0}^{m}p_kA^k$ and $q=\sum_{j}^{l}q_jA^j$ where we say that for $m<l$, we set $p_{m+1}=...=p_{m+l}=0$. The same goes for $l>m$.

The equation goes as follows: $$\sum_{k=0}^{m}p_kA^k\sum_{j}^{l}q_jA^j=\sum_{k=0}^{m}\sum_{j=0}^{l}p_kq_jA^{k+j}\overset{(1)}{=}\sum_{k=0}^{m+l}\sum_{j+l=k}p_kq_{j}A^k\overset{(2)}{=}\sum_{k=0}^{m+l}\sum_{j=0}^{k}p_jq_{k-j}A^k$$

I really do not get the $(1)$ and $(2)$ equality, or rather I do not know how to get there and whenever I tried something I do not even get to the $(1)$ . Any help is appreciated!

Bernard
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babemcnuggets
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    Take $m=1$ and $l=2$ and write everything out to see what happens. – Dietrich Burde Dec 11 '20 at 20:14
  • I see the (2) now, if I write the second expression out aswell as the last one, I see the equality, since the indices of $p_j$ and $q_k-j$ must sum up to $k$ which they do. So after organizig a bit one could get from the 2nd equality to the 4.th equality. However I still do not know how to understand the (1) in this equation. It is the solution but the sum makes no sense or am I simply not understanding it? – babemcnuggets Dec 11 '20 at 20:31

1 Answers1

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We obtain \begin{align*} \color{blue}{\sum_{k=0}^{m}p_kA^k\sum_{j=0}^{l}q_jA^j} &=\sum_{k=0}^{m}\sum_{j=0}^{l}p_kq_jA^{k+j}\\ &=\sum_{{0\leq k\leq m}\atop{0\leq j\leq l}}p_kq_jA^{k+j}\tag{1}\\ &=\sum_{n=0}^{m+l}\left(\sum_{{{0\leq k\leq m}\atop {0\leq j \leq l}}\atop {k+j=n}}p_kq_j\right)A^{n}\tag{2}\\ &\,\,\color{blue}{=\sum_{n=0}^{m+l}\left(\sum_{k=0}^{n}p_kq_{n-k}\right)A^{n}}\tag{3}\\ \end{align*}

Comment:

  • In (1) we use another convenient notation to write the double sum.

  • In (2) we rearrange the summands by increasing powers of $A^n$. Note the exponent $n$ of $A^n$ is coupled with the indices $k,j$ by explicitly writing $k+j=n$.

  • In (3) we eliminate the index $j$ by using $k+j=n$. Since $p_k=0$ if $n>m$ and $q_j=0$ if $n>l$ the representation of the inner sum in (3) is fine.

Markus Scheuer
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