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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 $k$?

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  • $\begingroup$ Due to the recurrence relations dlmf.nist.gov/10.51.i the function $I_k(\sigma)$ satisfies $(k+1)I_k(\sigma)+\sigma I_k'(\sigma)=\sigma I_{k-1}(\sigma)$. $\endgroup$ – Nemo Feb 13 '18 at 9:26
  • $\begingroup$ It seems that in general $I_k(\sigma)=R_1(\sigma)+R_2(\sigma){\mathrm{arctan}}\sigma$, where $R_{1,2}$ are rational functions. $\endgroup$ – Nemo Feb 13 '18 at 10:29
  • $\begingroup$ Note that for convergence of the Integral for $I_k$ one needs $l \geq k$. $\endgroup$ – Johannes Trost Feb 13 '18 at 11:37
  • $\begingroup$ @JohannesTrost no such relation is needed dlmf.nist.gov/10.53 $\endgroup$ – Nemo Feb 13 '18 at 11:57
  • $\begingroup$ Anyway I'm interested just in the case $l\ge k$. $\endgroup$ – Zurab Silagadze Feb 13 '18 at 12:04

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