How do I find the EGF of the sequence $a_n = s(s − 1)(s − 2). ... .(s − n + 1)$

Question

This is an exercise given in class, but I don't even know how to start. Can you give me some hints?

Answer 1

HINT:

$$a_n=s(s-1)(s-2)\ldots(s-n+1)=s^{\underline n}=\binom{s}nn!\;,$$

where $x^{\underline k}$ is the falling factorial. You want

$$\sum_{n\ge 0}a_n\frac{x^n}{n!}=\sum_{n\ge 0}\frac{a_n}{n!}x^n\;.$$