General factorization of $x^n-y^n$ (n is real)

Question

I am looking for a general expansion of $x^{n}-y^{n}$ with $x,y>0$ and $n$ being real. I came across the following formula (Proving $x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$)

$$x^n-y^n = (x-y)(x^{(n-1)}+x^{(n-2)}y+...+y^n)$$

Does it hold true for $n \in \mathbb{R}$?

I don't think there's a nice general factorization for $;n\in\Bbb R;$ . — DonAntonio, Jun 08 '17 at 18:27
It's not even clear what a "factorization" would mean in a case like this. — Thomas Andrews, Jun 08 '17 at 18:28
I wouldn't call this a general formula or a very useful factorization, but, for example, $x^e-y^e = (x^{\sqrt{e}} - y^{\sqrt {e}})(x^{\sqrt {e}} + y^{\sqrt {e}})$ — Χpẘ, Jun 08 '17 at 18:32
@Χpẘ That isn't correct - it is $x^e-y^e=(x^{e/2}-y^{e/2})(x^{e/2}+y^{e/2})$. — Thomas Andrews, Jun 08 '17 at 18:49
@user1110341 How can this question be a duplicate of the one that you mentioned? This question is concerned about the case in which $n$ is an arbitrary real number, whereas the one that you mentioned is about $n\in\Bbb N$. — José Carlos Santos, Aug 21 '23 at 08:13

score 2 · Answer 1 · answered Jun 08 '17 at 18:33

2

If you write $x = y + t$, then you can use the binomial series on $x^n = (y+t)^n$, so

$$\eqalign{ x^n - y^n &= \sum_{k=1}^\infty {n \choose k} y^{n-k} t^k\cr &=\sum_{k=1}^\infty {n \choose k} y^{n-k} (x-y)^k\cr &= (x-y) \sum_{k=1}^\infty {n \choose k} y^{n-k} (x-y)^{k-1}} $$ The series converges for $|x-y| < |y|$.

answered Jun 08 '17 at 18:33

Robert Israel

448,999

@{Robert} does it hold true for $n<0$? – Priyanshu Jun 08 '17 at 19:08
Yes, it works for $n < 0$, or for complex $n$ for that matter. – Robert Israel Jun 08 '17 at 19:24
Thanks @{Robert} – Priyanshu Jun 08 '17 at 21:19

AmagicalFishy · Answer 2 · 2017-06-08T18:50:40.197

0

You could factor out $x - y$, giving you:

$$(x - y)(x^{n-1} + x^{n-2}y + ... xy^{n - 2} + y^{n - 1}) = \\(x - y)\sum_{i=0}^{n-1}x^{i}~y^{n - 1 - i}$$

Edit: The question was changed to include this factorization after this answer was posted.

edited Jun 08 '17 at 18:50

answered Jun 08 '17 at 18:43

AmagicalFishy

1,950

AmagicalFishy, how many terms will the summation contain when $n \in R$ (say $n = 0.8$)? – Priyanshu Jun 08 '17 at 18:46

General factorization of $x^n-y^n$ (n is real)

2 Answers2