comment
I'm guessing Alex wants hypergeometric $$\begin{align}
{}_0F_1(;k+3/2;-t^2/4)&=1-\frac{t^2}{2(2k+3)}+\frac{t^4}{2\cdot 4\cdot(2k+3)\cdot(2k+5)}+\dots
\\
&=\Gamma(k+3/2) t^{-k-1/2} 2^{k+1/2} J_{k+1/2}(t)
\end{align}$$
inside the parentheses. Then, for example, with $k=10, q=3$ we get
$$
\int_0^1 \frac{2147483648\;\Gamma(23/2)^3 \sqrt{2} J_{21/2}(t)^3\,dt}{t^{63/2}}
= \frac{2043784548694122266058130317213\pi}{1869392021479944872009840721920}
$$
added
With $f=\Gamma(k+3/2) t^{-k-1/2} 2^{k+1/2} J_{k+1/2}(t)$, Maple 16 has these asymptotic claims as $k \to \infty$:
$$\begin{align}
\int_0^\infty f\,dt &= \sqrt{\pi}\,k^{1/2} + \frac{3\sqrt{\pi}}{8}\,k^{-1/2} - \frac{7\sqrt{\pi}}{128}\,k^{-3/2}+O(k^{-5/2})
\\
\int_0^\infty f^2\,dt &=\frac{\sqrt{\pi}}{\sqrt{2}}\,k^{1/2}+
\frac{9\sqrt{\pi}}{16\sqrt{2}}\,k^{-1/2}
-\frac{31\sqrt{\pi}}{512\sqrt{2}}\,k^{-3/2} + O(k^{-5/2})
\end{align}$$
and did not do $f^3$.