The spherical Bessel function closure relation is given by $$ \frac{2 r^2}{\pi} \int_0^\infty dk \; k^2 j_{L}(r k)j_L(r' k) = \delta(r - r') \ , $$ as given in Formula (4.1) in Mehrem, Londergan and Macfarlane's paper Analytic expressions for integrals of products of spherical Bessel functions. Do similar relations exist for the spherical Bessel functions $y_{L}(x)$? That is, formulas of the type $$ \int_0^\infty dk \; k^2 y_{L}(r k) y_L(r' k) = \ldots \ , $$ or cross-products like $$ \int_0^\infty dk \; k^2 j_{L}(r k) y_L(r' k) = \ldots \ ? $$
EDIT: I also found the above closure relation (1.17.14) on DLMF. Because of the identity $y_L(z) = (-1)^{L+1} j_{-L-1}(z)$ we have $j_{L}(z) = (-1)^{-L} y_{-L-1}(z)$ and it seems that we can rewrite the closure relation as $$ \frac{2 r^2}{\pi} \int_0^\infty dk \; k^2 (-1)^{-L} y_{-L-1}(r k) (-1)^{-L} y_{-L-1}(r' k) = \delta(r - r') \ , $$ and taking $-L -1 \to L$ the above is $$ \frac{2 r^2}{\pi} \int_0^\infty dk \; k^2 y_{L}(r k) y_{L}(r' k) = \delta(r - r') \ , $$ which I think makes sense as long as $L$ is allowed to be negative in the original closure relation I wrote above (which I think is true). Does this make sense? What about a closure relation for $y_{L}j_{L}$ combinations?