partial-differential-equations-by-evans · Problem 2

Four Important Linear PDE

Mathematics — Calculus · multiindex-notation

Problem

Prove the Multinomial Theorem

(x_1 + \ldots + x_n)^k = \sum_{|\alpha|=k} \binom{|\alpha|}{\alpha} x^\alpha,

where

\binom{|\alpha|}{\alpha} := \frac{|\alpha|!}{\alpha!}

\alpha! = \alpha_1! \alpha_2! \ldots \alpha_n!

, and

x^\alpha = x_1^{\alpha_1} \ldots x_n^{\alpha_n}

. The sum is taken over all multiindices

\alpha = (\alpha_1, \ldots, \alpha_n)

with

|\alpha| = k