Binomial expansion comparison with legendre polynomial
I've been working on this question which asks to show that
So first taking the n derivatives of the binomial expansions of (x2-1)n
and comparing it with
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
Have I made a mistake early on or is there a clever way to combine the two series?