I'm trying to find the closed form solution for the integral of the product of Bessel functions. Namely,
$$ I_{\alpha \beta} = \int_{0}^{\infty} dT e^{-2s T} J_{\alpha}(T) J_{\beta}(T) $$ where $s > 0$ is a real number and $\alpha, \beta$ are positive integers. Any help you would greatly appreciated! (Even the case where $\alpha = \beta$ would be good).
Identity 10.22.66 on DLMF states: $$\int_0^\infty e^{-a t}J_\nu(bt)J_\nu(ct)\mathrm{d}t=\frac{1}{\pi\sqrt{bc}}Q_{\nu-\frac{1}{2}}\left(\frac{a^2+b^2+c^2}{2bc}\right)$$ With $Q_n$ being the associated Legendre function of the second kind of order $n$. So in the case that $\alpha=\beta$, your integral should be $$\frac{1}{\pi}Q_{\beta-\frac{1}{2}}\left(2s^2+1\right)$$ I'm not sure what to do about the $\alpha\neq \beta $ case. Perhaps we can start with the smaller of the two and then make multiple uses of the recurrence $$J_{\nu+1}(z)=\frac{2\nu}{z}J_\nu(z)-J_{\nu-1}(z)$$ But I doubt this helps much.