The Bessel function of the first kind and order n has the integral representation
$J_n(z)=i^{-n}/\pi \int_0^\pi e^{iz\cos\theta}\cos(n\theta)d\theta$
By using the Laplace integral for the Legendre polynomial $P_n(x)$,
$P_n(x)=1/\pi\int_0^\pi (x+\sqrt{x^2-1}\cos\theta)^nd\theta$
Find the generating function
$\sum_{n=0}^{\infty} \frac{P_n(x)r^n}{n!}$
in terms of $J_0$.
Ok, so I've found $J_0=1/\pi \int_0^{\pi} e^{izcos\theta}d\theta$. That's not really a earth shattering amount of work but I can't find a way to relate the two together. From just plugging the legendre polynomials into the sum I get
1 + xr + 1/4(3x^2-1) + so on
but I still don't see any relation. Any help is appreciated.