By polynomial formula (sum_(i in m)x_i)^n=sum_(j_i in ZZ^+)(nj_0,…,j_(m−1))prod_(i in m)x^(j_i)_i

Jaxon Hamilton

Jaxon Hamilton

Answered question

2022-07-21

By polynomial formula
( i m x i ) n = j i Z + j i = n ( n j 0 , , j m 1 ) i m x i j i
where n N
But what about n N ?

Answer & Explanation

Marisa Colon

Marisa Colon

Beginner2022-07-22Added 18 answers

When n is not an integer, multivariate Taylor series expansion is to be applied. Let's expand ( a + b + c ) α around b = 0 and c = 0. We will need:
p + q b p c q ( a + b + c ) α = k = 0 p + q 1 ( α k ) ( a + b + c ) α p q = ( α p + q ) ( p + q ) ! ( a + b + c ) α p q
Thus:
( a + b + c ) α = p = 0 q = 0 ( α p + q ) ( p + q ) ! ( a ) α p q b p p ! c q q ! = p = 0 q = 0 ( α p , q , α p q ) a α p q b p c q
As you can see, in the case of α N , the series terminates, and the expansion reduces to the tri-nomial expansion.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?