In a quantum mechanical problem I encountered the integral $$I_k=\int_0^\infty x^{2(l+1)-k}j_k(\sigma x)e^{-x}[L_{n-l-1}^{2l+1}(x)]^2 dx,$$ where $j_k(x)$ is a spherical Bessel function, and $\sigma$ is a real parameter.

Using the following entry from Gradshtein and Ryzhik (p. 858) $$\int_0^\infty x^\alpha e^{-bx}L_n^\alpha(\lambda x) L_m^\alpha(\mu x)dx= \frac{\Gamma(m+n+\alpha+1)}{\Gamma(m+1)\;\Gamma(n+1)}\frac{(b-\lambda)^n(b-\mu)^m}{b^{m+n+\alpha+1}}\;F\left (-m,-n;-m-n-\alpha;\frac{b(b-\lambda-\mu)}{(b-\mu)(b-\lambda)}\right ),$$ and some properties of the hypergeometric function $F$, we can get $$I_0=\frac{2(n+l)!}{(n-l-1)!}\frac{\cos^{2l+4}(\phi)\;\sin{(2n\phi)}}{\sin{(2\phi)}}P_{\;n-l-1}^{(0,\;2l+1)}(\cos{(2\phi)}),$$ where $\tan{\phi}=\sigma$, and $P_{\;n-l-1}^{(0,\;2l+1)}(x)$ are Jacobi polynomials. Can this result be generalized for non-zero $k$?