There are various types of generating functions - definitions and examples are given below. Every sequence has a generating function of each type. The particular generating function that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.
Generating functions are often expressed in closed form as functions of a formal argument x. Sometimes a generating function is evaluated at a specific value of x. However, it must be remembered that generating functions are formal power series, and they will not necessarily converge for all values of x.
The ordinary generating function of a sequence an is
If an is the probability mass function of a discrete random variable, then its ordinary generating function is called a probability-generating function.
The ordinary generating function can be generalised to sequences with multiple indexes. For example, the ordinary generating function of a sequence anm (where n and m are natural numbers) is
The exponential generating function of a sequence an is
The Lambert series of a sequence an is
Dirichlet series are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence an is
Definitions
Ordinary generating function
When generating function is used without qualification, it is usually taken to mean an ordinary generating function.Exponential generating function
Lambert series
Note that in a Lambert series the index n starts at 1, not at 0.
Dirichlet series generating functions
Dirichlet series generating functions are especially useful for multiplicative functions, when they have an Euler product expression. If an is a Dirichlet character then its Dirichlet series generating function is called a Dirichlet L-series.
Generating functions for the sequence of square numbers an = n2 are :-
Examples
Ordinary generating function
Exponential generating function
Dirichlet series generating function
Uses
Generating functions are used to :-See also
References
External Links