Suppose that the generating function is represented by an infinite continued fraction of the form
and that denotes the th convergent to this continued fraction expansion defined such that for all . Then:
# the function is rational for all where we assume that one of divisibility criteria of is met, that is, for some ; and
# if the integer divides the product , then we have . (en)
A generating function is a clothesline on which we hang up a sequence of numbers for display. (en)
A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag. (en)
