I've been working on this question which asks to show that

{{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}  {d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}}

So first taking the n derivatives of the binomial expansions of (x2-1)n


\frac{{{d}^{n}}}{d{{x}^{n}}}...=\sum\limits_{k=0}^  {n}{{{(-1)}^{k}}\frac{n!}{k!(n-k)!}(2n-2k)(2n-2k-1)...(2n-2k-n+1){{x}^{2n-2k}}}

and comparing it with


=\frac{1}{{{2}^{n}}}\sum\limits_{m=0}^{\frac{n}{2}  }{{{(-1)}^{m}}\frac{(2n-2m)!}{m!(n-m)!(n-2m)!}{{x}^{n-2m}}}

I'm having trouble with the final part,

It's clear that there's a factor of 1/n!2n difference between them but also

the Pn(x) series has m=0...n/2, and also xn , where as the n'th derivative series has k=0...n and x2n.

How can you rewrite one in terms of the other so they both have the same sum limits?

I've tried setting k=2s in the n'th derivative series and a bunch of other similar changes, but non will change the n'th powers of x.

The reason I noticed this was because the last terms of the series arn't the same,

the first series has last term, (-n)! on the bottom, means 1/infinity right?


and the second

\frac{1}{{{2}^{n}}}{{(-1)}^{\frac{n}{2}}}\frac{n!}{n!(\frac{n}{2})!0!}{{x  }^{0}}

Have I made a mistake early on or is there a clever way to combine the two series?