Generating function transformation

Article summary

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 (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations). 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 = f_0 + f_1 z + f_2 z^2 + \cdots : \widehat{F}(z) = \sum_{n=0}^\infty \frac{f_n}{n!} z^n = \frac{f_0}{0!} + \frac{f_1}{1!} z + \frac{f_2}{2!} z^2 + \cdots. In this article, we use the convention that the ordinary (exponential) generating function for a sequence \{f_n\} is denoted by the uppercase function F(z) / \widehat{F}(z) for some fixed or formal 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 [z^n]F(z) := f_n . The main article gives examples of generating functions for many sequences.…

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