Why Can I divide generating function by $x$

Question

In many books on generating functions author performs following operation to shift coefficients of $F(x) = \sum_i f_ix^i$ to the left $${F(x) - f_0} \over x$$ which in can be written as $$(F(x) - f_0) \times {1 \over x}$$ I do not understand why they can do that if generating function $x$ does not have any reciprocal so expression like $1 \over x$ does not make sense for generating function. Yet almost every author uses this trick.

Answer 1

Generating functions (with real coefficients) live in the ring $\mathbb{R}[[x]]$ of formal power series over $\mathbb{R}$ (see http://en.wikipedia.org/wiki/Formal_power_series). You are quite right that $x$ is not invertible in this ring (its group of units comprises precisely the series with non-zero constant term). However $\mathbb{R}[[x]]$ embeds in the field $\mathbb{R}((x))$ of formal Laurent series over $\mathbb{R}$ (see http://en.wikipedia.org/wiki/Formal_power_series#Formal_Laurent_series), in which $x$ is invertible. So the apparent abuse of notation can be made rigorous by stating that you are working in $\mathbb{R}((x))$.

Answer 2

\begin{equation} F(x) = f_0 + f_1x + f_2x^2 + \ldots \end{equation} Therefore we get that \begin{equation} F(x)-f_0 = f_1x + f_2x^2 + \ldots = x(f_1 + f_2x + \ldots) \end{equation} As you can see this expression is divisible by $x$.