I wish to thank in advance anyone willing to take a peek at my question. I'm facing a problem involving the following integral: $$f(y)=\int_0^\infty J_{n}(xy) _{2}F_{2}(\frac{n+1}{2}, \frac{n+2}{2}; \frac{n+3}{2}, n+1;e^{-ax^2}) x^{n+2} dx $$
where $_{2}F_{2}(\frac{n+1}{2}, \frac{n+2}{2}; \frac{n+3}{2}, n+1;e^{-ax^2})$ is the generalized hypergeometric function and $J_{n}(x) $ is the n-th order Bessel function of the first kind. This is in fact the Hankel transform of
$$ _{2}F_{2}(\frac{n+1}{2}, \frac{n+2}{2}; \frac{n+3}{2}, n+1;e^{-ax^2})x^{n+1} $$
(or in other words the radial part of a Fourier transform in polar coordinates). I know that the aforementioned function $_{2}F_{2}$ asymptotically approaches zero for large values of x and all possible orders n (as does $J_{n}(x) $), but on the other hand the integral itself does not seem to converge for all possible orders of n.
I think this fact may stem from the definition of the generalized hypergeometric function as a power series, but I still cannot quite explain it to myself. Could someone perhaps offer a more detailed/correct explanation?
(As far as evalution of such integrals is concerned, I'm not attempting to obtain an analytical solution, but rather evaluate the integral numerically)