Page "Generating function" Paragraph 1

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.

The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.