How to calculate the integral of a Legendre polynomial I would like to show that
$$
\int_{0}^{1}P_{l}(1-2u^{2})e^{2i\alpha u}du=i\alpha j_{l}(\alpha)h_{l}(\alpha)
$$
where $P_{l}(x)$ are the Legendre polynomials, $\alpha$ is a positive constant and $j_{l}$ and $h_{l}$ are the spherical Bessel and Hankel functions of the first kind, respectively.
I can use an expansion like
$$
P_{l}(1-2u^2)=\sum_{k=0}^{l}{l\choose k}{-l-1\choose k}u^{2k}
$$
but this leads to a solution as a sum of incomplete Gamma functions, so I'm a bit stuck.
Thanks in advance for any help.
 A: Assuming $0<u<v$ and $w=\sqrt{u^2+v^2-2uv\cos\theta}$, combining the Gegenbauer expansions for half-integer orders here and here gives
\begin{align}
 \frac{\exp iw}{w}&=\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)(-\mathsf{y}_{n}\left(u\right)+i\mathsf{j}_{n}\left(u\right))P_{n}\left(\cos\theta\right)\\
 &=i\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf{h}_{n}^{(1)}\left(u\right)P_{n}\left(\cos\theta\right)
\end{align}
The orthogonality of the Legendre polynomials reads
\begin{equation}
 \int_{-1}^1P_n(x)P_{\ell}(x)\,dx=\frac{2}{2\ell+1}\delta_{\ell n}
\end{equation}
or
\begin{equation}
 \int_0^\pi P_n(\cos\theta)P_{\ell}(\cos\theta)\sin\theta\,d\theta=\frac{2}{2\ell+1}\delta_{\ell n}
\end{equation}
The obtained identity is now projected on $P_\ell(\cos\theta)$ after multiplication by $\sin\theta$:
\begin{align}
 \int_{0}^\pi \frac{\exp iw}{w}P_l(\cos\theta)\sin\theta\,d\theta=i\mathsf{j}_{\ell}\left(v\right)\mathsf{h}_{\ell}^{(1)}\left(u\right)
\end{align}
(only the term $n=\ell$ survives in the summation). Both sides of the above identity are continuous functions of $u$ and $v$, the obtained expression remains valid for $u=v$. Choosing $u=v=\alpha$, it writes
\begin{equation}
  \int_{-1}^1 \frac{\exp i\alpha\sqrt{2}\sqrt{1-\cos\theta}}{\alpha\sqrt{2}\sqrt{1-\cos\theta}}P_l(\cos\theta)\sin\theta\,d\theta=2i\mathsf{j}_{\ell}\left(\alpha\right)\mathsf{h}_{\ell}^{(1)}\left(\alpha\right)
\end{equation}
Now, changing $\cos\theta=1-2t^2$ results in the following expression
\begin{equation}
 \int_0^1 e^{2i\alpha t}P_\ell(1-2t^2)\,dt=i\alpha\mathsf{j}_{\ell}\left(\alpha\right)\mathsf{h}_{\ell}^{(1)}\left(\alpha\right)
\end{equation}
which is identical to the proposed identity.
