Definition:Exponential Generating Function

Definition

Let $A = \left \langle {a_n}\right \rangle$ be a sequence in $\R$.

Then $\displaystyle G_A \left({z}\right) = \sum_{n \mathop \ge 0} \frac {a_n} {n!} z^n$ is called the (exponential) generating function for the sequence $A$.

The mapping $G_A \left({z}\right)$ is defined for all $z$ for which the power series $\displaystyle \sum_{n \mathop \ge 0} \frac {a_n} {n!} z^n$ is convergent.

The definition can be modified so that the lower limit of the summation is $b$ where $b > 0$ by assigning $a_k = 0$ where $0 \le k < b$.

Also see

• Definition:Generating Function
• Power Series Expansion for Exponential Function

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 4.1$: Generating functions
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions