Relationship between Legendre polynomials and Hypergeometric functions

[Rulonegger]

Homework Statement

If we define \xi=\mu+\sqrt{\mu^2-1}, show that
P_{n}(\mu)=\frac{\Gamma(n+\frac{1}{2})}{n!\Gamma(\frac{1}{2})}\xi^{n}\: _2F_1(\frac{1}{2},-n;\frac{1}{2}-n;\xi^{-2}) where P_n is the n-th Legendre polynomial, and _2F_1(a,b;c;x) is the ordinary hypergeometric function.

Homework Equations

\frac{1}{\sqrt{1-2\mu t+t^2}}=\sum_{n=0}^{\infty}{t^n P_{n}(\mu)}
_2F_1(a,b;c;x)=\sum_{n=0}^{\infty}{\frac{(a)_n (b)_n}{(c)_n}\frac{x^n}{n!}}
(\alpha)_n=\alpha(\alpha+1) _\cdots (\alpha+n-1)

The Attempt at a Solution

I just tried to write down how _2F_1(\frac{1}{2},-n;\frac{1}{2}-n;\xi^{-2}) is, and expand \xi^{-2} with the binomial theorem in terms of \mu, but it results in a little complicated double infinite sum, so i feel that there is another way to prove it, but i cannot find it.

 

[andrien]

It will be better if you deal with eqn itself.Just try to convert hypergeometric differential eqn to legendre one by change of variable.Also see what those a,b and c are by comparison.make the change as t=1/2(1-u),u is of legendre and t for hypergeometric.