Let $n$ be half an odd integer, say $n=k+1/2, k \in \mathbb{N}$.
Let $q\geq 1$. I would like to calculate (or approximate) the following integral: $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot 1\cdot 3\cdot 5\cdots (2k+1) \frac{J_{k+\frac 12}(t)}{t^{k+ \frac 12}}\right)^q t\ dt. $$
Any ideas or references will be very helpful.
Thank you.