I need to solve the following integral:
$$\int_0^1 j_\ell(z_{n\ell}x)\:j_{\ell'}(z_{n'\ell'}x)\:x^4 \text{d}x$$
where $j_\ell$ is the $\ell$-th spherical Bessel function of the first kind, and $z_{n\ell}$ is the $n$-th root of $j_\ell$.
I'm looking for a source that can indicate a way to calculate these integrals. I cannot find them in the usual online sources (Wikipedia, Mathematica,etc).
The problem with the spherical Bessel integral is that it needs a recursive relation between the $j_\ell(z_{n\ell}x)$ and $_\ell(z_{n'\ell}x$ which is not at all trivial. Any thoughts?
The spherical Bessel function can be defined from the Bessel function as ([1], see equation 1):
\begin{equation} j_{n}(z)=\sqrt{\dfrac{\pi}{2z}}J_{n+\frac{1}{2}}(z) \end{equation}
I think that from this equation one can establish a recurrence relation using the original Bessel functions. Additionally, we have the following result ([1], see equation 5):
\begin{equation} \int_0^{2\pi}\int_0^\pi\: Y_\ell^{m}(\varphi,\theta) Y_{\ell'}^{*m'}(\varphi,\theta)\sin(\varphi)\text{d}\varphi\text{d}\theta=\delta_{ll}\delta{mm} \end{equation}
References
[1] Baddour, N. (2010). Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates. JOSA A, 27(10), 2144-2155.