I am taking some time to carefully construct my understanding of generating functions as they are extremely interesting to me in terms of properties and uses.
I am making my way to convolutions now and the formula goes:
For sequences a = {$a_i:i\ge0$} and b = {$b_i:i\ge0$} we have that their convolution $c$ is $$c_n = a_0b_n+a_1b_{n-1}+…+a_nb_0$$ $$=\sum_{i=0}^na_ib_{n-i}$$
My question is why we choose to define the convolution by pairing the $(i)th$ moment of $a$ with the $(n-i)th$ moment of $b$ as opposed to equating the indices for both. What information is being captured by having the sequences run in opposite directions? I tend to come at this from relating the information to probability theory but any approach is welcome.