(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Using binomial expansion, prove that

[tex]

\frac{1}{\sqrt{1 - 2 x u + u^2}} = \sum_{k} P_k(x) u^k.

[/tex]

2. Relevant equations

[tex]

\frac{1}{\sqrt{1 + v}} = \sum_{k} (-1)^k \frac{(2k)!}{2^{2k} (k!)^2} v^k

[/tex]

3. The attempt at a solution

I simply inserted [itex]v = u^2 - 2 x u[/itex], then expanded the [itex]v^k[/tex] to obtain the double sum

[tex]

\sum_{k} (-1)^k \frac{(2k)!}{2^{2k} (k!)^2} \sum_{n \leq k} \left( \begin{array}{c} k \\ n \end{array} \right) (-2 x)^n u^{2 k - n}.

[/tex]

Now I need to turn this into a single sum by collecting like powers of [itex]u[/itex], which is what I'm stuck at. I don't see how to go about that.

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# Generating function for Legendre polynomials

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