How to relate $\int_{-1}^{1}e^{ikru} P_{\ell}(u)du$ to $j_\ell(kr)$ in a simple way? Consider the following epansion of the function  $e^{ikru}$ in terms of Legendre polynomials, $P_\ell(u)$, $$e^{ikru}=\sum_{\ell=0}^{\infty}C_\ell(r)P_\ell(u)$$ where $k$ is a constant real parameter, and $r,u$ are a real variables with $0\leq r<\infty$ and $-1\leq u\leq 1$. Using the orthonormality of Legendre polynomials, $$\int_{-1}^{1} P_\ell(u)P_{\ell'}(u)du=\frac{2}{2\ell+1}\delta_{\ell\ell'}$$ we can show,
$$C_\ell(r)=\frac{2\ell+1}{2}\int_{-1}^{1}e^{ikru} P_{\ell}(u)du\\
\hspace{3.5cm}  =\frac{2\ell+1}{2}\int_{-1}^{1}\left(\sum_{n=0}^{\infty}\frac{(ikru)^n}{n!}\right) P_{\ell}(u)du$$
How can we relate the integral on the right-hand side to the spherical Bessel functions $j_\ell(kr)$ in an efficient way?
 A: From the Rodrigues formula for the Legendre polynomials
\begin{equation}
 P_\ell(u)=\frac{(-1)^\ell}{2^\ell \ell!}\frac{d^\ell}{du^\ell}\left[(1-u^2]^\ell\right]
\end{equation}
we have
\begin{align}
 C_\ell (kr)&=\frac{2\ell+1}{2}\int_{-1}^{1}e^{ikru} P_{\ell}(u)\,du\\
 &=\frac{(-1)^\ell(2\ell+1)}{2^{\ell+1} \ell!}\int_{-1}^{1}e^{ikru} \frac{d^\ell}{du^\ell}\left[(1-u^2]^\ell\right]\,du\\
\end{align}
Remarking that
\begin{equation}
 \left.\frac{d^p}{du^p}\left[(1-u^2]^\ell\right]\,du\right|_{u=\pm1}=0
\end{equation}
for $0<p<\ell-1$, performing $\ell$ successive integrations by parts gives
\begin{align}
 C_\ell (kr)&=\frac{(-1)^\ell(2\ell+1)}{2^{\ell+1} \ell!}(ikr)^\ell(-1)^\ell\int_{-1}^{1}e^{ikru} (1-u^2)^\ell\,du\\
 &=\frac{2\ell+1}{2^{\ell+1} \ell!}(ikr)^\ell\int_{-1}^{1}\cos(kru) (1-u^2)^\ell\,du
\end{align}
(using a parity property). Changing $u=\cos\theta$,
\begin{equation}
 C_\ell (kr)=\frac{2\ell+1}{2^{\ell+1} \ell!}(ikr)^\ell\int_0^{\pi}\cos(kr\cos\theta)(\sin\theta)^{2\ell+1}\,d\theta
\end{equation}
Now, from the integral representation of the spherical Bessel function
\begin{equation}
 j_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin\theta)^{2n+1}\,d\theta
\end{equation}
we obtain
\begin{equation}
 C_\ell (kr)=(2\ell+1)i^\ell j_\ell(kr)
\end{equation}
(Alternatively, the Bessel function can be recognized by expanding $\cos(kr\cos\theta)$ in powers of $kr$ and by using Euler's Beta integral. The series expansion of $j_\ell(kr)$ is then obtained).
