Exponential generating function

Definition

Let $K$ be a field of characteristic $0$ , for example $\mathbb{Q}$ or $\mathbb{R}$ , and let $(a_n)_{n \ge 0}$ be a sequence in $K$ . The exponential generating function of $(a_n)$ is the formal power series $\hat{A}(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!} \in K[\![x]\!].$

Remarks

The constant sequence $1$ has the exponential generating function $e^x = \sum_n x^n / n!$ . For exponential generating functions, the product corresponds to the binomial convolution $c_n = \sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$ . They suit labelled structures. For example, the derangement numbers have the exponential generating function $\frac{e^{-x}}{1 - x}$ .

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Formal power series definition
Exponential generating function definition

Tag	`0KE6NBTVY8`
URI	`m2://0KE6NBTVY8`
Kind	definition
Writing	W1 stated
MSC	05A15
Defines	Exponential generating function
Uses	Formal power series

Exponential generating function

Definition

Remarks

Reading path

Further reading

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