Editing Generating function (section)

{{Short description|Formal power series; coefficients encode information about a sequence indexed by natural numbers}}
{{About|generating functions in mathematics|generating functions in classical mechanics|Generating function (physics)|generators in computer programming|Generator (computer programming)|the moment generating function in statistics|Moment generating function}}

In [[mathematics]], a '''generating function''' is a representation of an [[infinite sequence]] of numbers as the [[coefficient]]s of a [[formal power series]]. Generating functions are often expressed in [[Closed-form expression|closed form]] (rather than as a series), by some expression involving operations on the formal series. 

There are various types of generating functions, including '''ordinary generating functions''', '''exponential generating functions''', '''Lambert series''', '''Bell series''', and '''Dirichlet series'''. 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.

Generating functions are sometimes called '''generating series''',<ref>This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", ''[[Canadian Journal of Mathematics]]'' 3, [https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407 p.&nbsp;405–411], but its use is rare before the year 2000; since then it appears to be increasing.</ref> in that a series of terms can be said to be the generator of its sequence of term coefficients.