Generating function transformation

This article is about generating functions in mathematics. For other uses, see Generating function transformation (disambiguation).

Definitions

Given a sequence $\{f_{n}\}_{n=0}^{\infty }$, the ordinary generating function (OGF) of the sequence, denoted $F(z)$, and the exponential generating function (EGF) of the sequence, denoted ${\widehat {F}}(z)$, are defined by the formal power series

$F(z)=\sum _{n=0}^{\infty }f_{n}z^{n}$

${\widehat {F}}(z)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n!}}z^{n}.$

The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions (DGFs), Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful formulas.