$$\frac{1}{(1-x^a)(1-x^b)}=\frac{A}{(1-x)^2}+\frac{B}{(1-x)}+\sum_{r^a=1}^{ }\frac{C_r}{(1-x/r)}+\sum_{t^b=1}^{ }\frac{D_t}{(1-x/t)}$$ $(t,r\neq1; A, B, C, D$ are real numbers)
How did the author know that he need to separate the case r,t=1?
I would appreciate it really much if someone can guide me to resources to study the general rules for partial fraction expansion like this.
This expansion is only valid if $a$ and $b$ are coprime. In that case, $t=r=1$ is the only multiple root of the denominator. For a root $x_0$ of the denominator with multiplicity $m$, a partial fraction expansion contains terms proportional to $(x-x_0)^{-k}$ for all $k$ up to $m$. If $t=r=1$ hadn’t been treated separately, both sums would have included a term proportional to $(1-x)^{-1}$, and the term proportional to $(1-x)^{-2}$ would have been missing. If $a$ and $b$ aren’t coprime, there are further multiple roots that need to be treated analogously.