1 Answers
In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function or weighted sums over the higher-order derivatives of these functions.
Given a sequence, { f n } n = 0 ∞ {\displaystyle \{f_{n}\}_{n=0}^{\infty }} , the ordinary generating function of the sequence, denoted F {\displaystyle F} , and the exponential generating function of the sequence, denoted F ^ {\displaystyle {\widehat {F}}} , are defined by the formal power series
In this article, we use the convention that the ordinary generating function for a sequence { f n } {\displaystyle \{f_{n}\}} is denoted by the uppercase function F {\displaystyle F} / F ^ {\displaystyle {\widehat {F}}} for some fixed or formal z {\displaystyle z} when the context of this notation is clear. Additionally, we use the bracket notation for coefficient extraction from the Concrete Mathematics reference which is given by F := f n {\displaystyle F:=f_{n}}.The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet generating functions , Lambert series, and Newton series. In this article we focus on transformations of generating functions in mathematics and keep a running list of useful transformations and transformation formulas.