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There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below.Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably.Generating functions are not functions in the formal sense of a mapping from a domain to a codomain.Generating functions are sometimes called generating series, 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.
For example, the ordinary generating function of a two-dimensional array a.
These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x.
Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its series expansion; this explains the designation "generating functions".
The main article provides several more classical, or at least well-known examples related to special arithmetic functions in number theory.
Note that in a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined.
The sum of this infinite series is the generating function.