Partial fraction decomposition of $\frac{1}{x^a(x+c)^b}$ We have the partial fraction decomposition
$$\frac{1}{x^a(x+c)^b}=\frac{d_a}{x^a}+\frac{d_{a-1}}{x^{a-1}}+...+\frac{d_{1}}{x}+\frac{e_b}{(x+c)^b}+\frac{e_{b-1}}{(x+c)^{b-1}}+...+\frac{e_{1}}{x+c},$$
where $a,b,c\in\mathbb{N}$.
Is there anything particular we can say about the nature of coefficients $d_i$ and $e_i$? Any references would be greatly appreciated.
P.S. Sorry for being so open-ended.

Multiply both sides by $x^a$ and $(x+ c)^b$ to get $$1= d_a(x+ c)^b+ d_{a-1}x(x+ c)^b+ \cdots+ d_1x^{a-1}(x+ c)^b+ e_bx^a+ e_{b+1}x^a(x+ c)+ \cdots+e_1x^a(x+ c)^{b-1}~.$$
Taking $~x= 0~$ gives immediately $~1= c^b~d_a~$ so $~d_a= 1/c^b~$.  Taking $~x= -c~$ immediately gives $~1= e_b~c^a~$ so $~e_b= 1/c^a~$.
That leaves $~a-1+ b-1= a+ b- 2~$ values to be determined.  Although there are no more "easy" equations but taking $~a+ b- 2~$ different values for $~x~$ will give $~a+ b- 2~$ equations for the $~a+ b- 2~$ unknown values.
