I am aware of the the closure relation for spherical Bessel functions $$ \int_0^\infty{dx\; x^2 \; j_l(k_1x)j_l(k_2x)} = \frac{\pi}{2k^2} \delta(k_1 - k_2) $$ as well as the orthogonality relation. I would love to know if it is possible to solve integrals with higher powers of $x$. Specifically, what is the solution of $$ \int_0^\infty{dx\; x^6 \; j_l(k_1x)j_l(k_2x)} . $$ I found a recursion relation in a paper from 2017 but the resulting expression is insanly complex. I'm afraid I'm pretty clueless in this area... Sorry.
2026-03-26 19:00:39.1774551639
Is there an equivalent for closure relation for spherical Bessel function for the 6th power of $x$?
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