The Dirac-$\delta$ has the following representation using Bessel functions (perhaps best interpreted as a resolution of the identity for the Hankel transform / Fourier transform in polar coordinates):
(1) $\quad \delta(b - a) = a \, \int_{\mathbb{R}_{\geq 0}} J_n(r \, a) \, J_n(r \, b) \, r \, dr$
where the $J_n(\cdot)$ are (cylindrical) Bessel functions and $a,b>0$, $n > -1$, see e.g. (Y. T. Li and R. Wong, “Integral and series representations of the dirac delta function,” Commun. Pure Appl. Anal., vol. 7, no. 2, pp. 229–247, Dec. 2008).
In a calculation I obtain a generalization of the above integral,
(2) $\quad \int_{\mathbb{R}_{\geq 0}} J_{n+1}(r \, a) \, J_n(r \, b) \, r \, dr$
which I try to make sense of. Eq. (1) suggest to interpret it in a weak sense, i.e.
$\int_{\mathbb{R}_{\geq 0}} \phi(a) \Big( \int_{\mathbb{R}_{\geq 0}} J_{n+1}(r \, a) \, J_n(r \, b) \, r \, dr \Big) a \, da$
where $\phi(x)$ is some test function. When $\phi(a)$ is a Gaussian then an analytic solution exists and one sees that the integral over $r$ yields a kernel that (slightly) smoothes $\phi(a)$ with some weak dependence on $n$.
I am looking for a general result in this direction, e.g. of Eq. (2) as a Dirac-$\delta$ plus some smooth kernel. Computing Eq. (2) one obtains a singularity at $a = b$ and this is my main obstacle in making sense of it.