Assuming that the product is associative, I would like to show that
$$ \sum_{s=0}^{k} (\sum_{i=0}^{s} (\alpha_i \cdot \beta_{s-i}) \cdot \gamma_{k-s}) = \sum_{s=0}^{k} ( \sum_{i=0}^{k-s} \alpha_s \cdot (\beta_i \cdot \gamma_{k-s-i})) $$
If I write the expressions out, it is clear that they are but is there a way to do this in a more elegant way ? I tried a change of variables but didn't succeed. Any help would be appreciated :)
$$\begin{align*} \sum_{s=0}^k\left(\sum_{i=0}^s(\alpha_i\cdot\beta_{s-i})\cdot\gamma_{k-s}\right)&=\sum_{{0\le i,j,\ell}\atop{i+j+\ell=k}}(\alpha_i\cdot\beta_j\cdot\gamma_\ell)\\ &=\sum_{s=0}^k\left(\sum_{i=0}^{k-s}\alpha_k\cdot(\beta_i\cdot\gamma_{k-s-i}\right) \end{align*}$$