# Generating function by Partial fraction expansion

Suppose then that the generating function is of the form: $P(s)=U(s)/V(s)$. Where $U$ and $V$ are polynomials without common roots. For simplicity let us first assume that the degree of $U$ is lower than the degree of $V$, say $m$. Moreover, suppose that the equation $V(s) = 0$ has $m$ distinct roots $s_1, s_2, ... ,s_m$. Then $V(s)=(s-s_1)(s-s_2)...(s-s_m)$.

It can be decomposed into partial fractions $P(s)=\frac{z_1}{(s_1-s)}+...+\frac{z_m}{(s_m-s)}$. Where $z_1,...z_m$ are constants. Multiply by $s_1-s$ as $s->s_1$ then $(s_1-s)P(s)=\frac{-U_s}{(s-s_2)(s-s_3)...(s-s_m)}$. As $s->s_1$ the denomiator tends to $V'(s_1)$.

Why the denominator tends to $V'(s_1)$? I can't see it.

## 1 Answer

The definition of the derivative at a point $s_1$ is

$$\lim_{x\rightarrow s_1}\frac{V(x)-V(s_1)}{x-s_1}$$

• Um... Yes and what? – Sargis Iskandaryan Feb 15 '18 at 14:41
• Holy Molly! Thank you )) – Sargis Iskandaryan Feb 15 '18 at 23:53
• Glad it clicked :) – law-of-fives Feb 16 '18 at 0:43