I am able to understand this text except for the side note where they have mentioned:
To be precise equation $12.8$ holds for $n\ge1$ and equation $12.9$ holds for $n\ge2$. But since equation 12.8 is $0$ for $n=1$ and equation 12.9 is $0$ for $n=1,2$, both equations hold for all $n\ge0$.
I can understand that equation $12.8$ holds for $n\ge1$ as we have derived it to calculate the coefficient of $x^{n-1}$ and if if we put $n=0$, we'll be calculating the coefficient of $x^{-1}$ which is wrong. Can anyone one confirm if this explanation is correct?
And I can't understand the second part of this side note at all, i.e., But since equation 12.8 is $0$ for $n=1$ and equation 12.9 is $0$ for $n=1,2$, both equations hold for all $n\ge0$.