Guichard-Combinatorics-Graph-Theory · Problem 1.8.5

Fundamentals

Mathematics — Probability · stirling-numbers

Problem

Show that

x^{\underline{n}} = \prod_{k=0}^{n-1}(x-k) = \sum_{i=0}^{n} s(n,i) x^i

n \geq 1

;

x^{\underline{n}}

is called a falling factorial. Find a similar identity for

x^{\overline{n}} = \prod_{k=0}^{n-1}(x+k)

;

x^{\overline{n}}

is a rising factorial.