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.