Delta-function representations, Bessel function I know there is a $\delta$-function representation in terms of the Bessel function as
$$\delta(x-a) = x\int_0^{\infty} t J_{\nu}(xt)J_{\nu}(at)\; dt$$
Is there something similar for
$$\int_0^{\infty} t J_{\mu}(xt)J_{\nu}(at)\; dt$$
Maybe there is something I can do to at least get a reasonable series representation... All I can find are the integrals where the power of $t$ is related in a certain way to the orders $\mu$ and $\nu$, but it is not the case in my case. Thanks!

In general, $\mu$ and $\nu$ are not equal, representation in the form of series can be obtained if you replace the product of only Bessel function under the integral by expansion in series where each term is expressed via hypergeometric function $_2F_1$ (see Bateman and Erdelyi, Higher Transcedental Functions, Volume $2$, Chapter $7 (7.2.7$). Then you can integrate term by term.
