Combinatorics and Graph Theory · Problem 1.8.6

Fundamentals — David Guichard

Mathematics — Combinatorics · stirling-numbers-second-kind

Problem

Show that

\displaystyle\sum_{k=0}^{n} \left\{\begin{smallmatrix}n\\k\end{smallmatrix}\right\} x^{\underline{k}} = x^n

n \geq 1

;

x^{\underline{k}}

is defined in the previous exercise. The previous exercise shows how to express the falling factorial in terms of powers of

x

; this exercise shows how to express the powers of

x

in terms of falling factorials.