Partial fraction expansion of generating functions (clarification of a proof)

Question

I give below a part of Feller's. I am struggling to understand how equation 4.8 was derived. Any help will be much appreciated! Thanks

Answer 1

The equation (4.3) tells us that $$P(s)=\sum_{i=1}^m \frac{\rho_i}{s_i-s}.$$

If we substitute (4.6) and (4.7), we find that $$P(s)=\sum_{i=1}^m \frac{\rho_i}{s_i}\cdot\frac{1}{1-\frac{s}{s_i}} = \sum_{i=1}^m \frac{\rho_i}{s_i}\cdot(1+\frac{s}{s_i}+(\frac{s}{s_i})^2+(\frac{s}{s_i})^3+\ldots).$$

We can rearrange this formula in terms of powers of $s$: $$\begin{eqnarray*}P(s) &=& \sum_{i=1}^m \frac{\rho_i}{s_i}(\cdot1) + \sum_{i=1}^m \frac{\rho_i}{s_i}\cdot\frac{s}{s_i} + \sum_{i=1}^m \frac{\rho_i}{s_i}\cdot (\frac{s}{s_i})^2+\ldots\\&=& \sum_{i=1}^m \frac{\rho_i}{s_i} + \sum_{i=1}^m \frac{\rho_i}{s_i^2}\cdot s + \sum_{i=1}^m \frac{\rho_i}{s_i^3}\cdot s^2 + \ldots \end{eqnarray*}$$

Since each $p_n$ is the coefficient of $s^n$ in this expression, we see that indeed $p_n$ is of the form given in (4.8): $$p_n=\frac{\rho_1}{s_1^{n+1}}+\ldots+\frac{\rho_m}{s_m^{n+1}}.$$