I need help finding a generating function using some relation between the Bessel function and the Laplace integral for the Legendre Polynomials 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. 
 A: Define the generating function:
$$G \left( x,r \right) =\sum _{n=0}^{\infty }{\frac {P_{{n}} \left( x
 \right) {r}^{n}}{n!}},\tag{1}$$
and substitute the integral representation of the Legendre polynomials:
$$P_{{n}} \left( x \right) ={\frac {\int _{0}^{\pi }\! \left( x+\sqrt {{
x}^{2}-1}\cos \left( \theta \right)  \right) ^{n}{d\theta}}{\pi }},\tag{2}$$
into $(1)$ to obtain:
\begin{aligned}
G \left( x,r \right) &=\sum _{n=0}^{\infty } \frac {\left(\int _{0}^{\pi }\! \left( x+\sqrt {{x}^{2}-1}\cos \left( \theta \right)  \right) ^{n}{d\theta}\right){r}^{n}}{\pi n! },\\
&=\frac{1}{\pi }\int _{0}^{\pi }\! \sum _{n=0}^{\infty } \frac {\left( x+\sqrt {{x}^{2}-1}\cos \left( \theta \right)  \right) ^nr^n}{n! }{d\theta},\\
&=\frac{1}{\pi }\int _{0}^{\pi }\! \text{exp}\left[ rx+r\sqrt {{x}^{2}-1}\cos \left( \theta  \right)\right] {d\theta},\\
&=\frac{e^{rx}}{\pi }\int _{0}^{\pi }\! \text{exp}\left[r\sqrt {{x}^{2}-1}\cos \left( \theta  \right)\right] {d\theta},\\
&=e^{rx}J_0\left(i r \sqrt{x^2-1}\right)=e^{rx}I_0\left(r \sqrt{x^2-1}\right).
\end{aligned}
To move from line 1 to line 2 in $(3)$ I have switched the order of summation and integration, this usually requires justification (show convergence of integral and sum e.t.c..) but I will leave that to you. To move from the penultimate line to the last I used the integral representation of the Bessel function and solved for $iz$; I dropped a minus sign in the argument of the Bessel function as the function is even.
As a check you can Taylor expand the final expression in $r$ and you will find that the coefficients are indeed given by $P_n(x)/{n!}$.
