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Let $n$ be half an odd integer, say $n=k+1/2, k \in Z$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot 1\cdot 3\cdot 5\cdot \ldots \cdot (2k+1) \frac{J_{k+\frac 12}(t)}{t^{k+ \frac 12}}\right)^q t dt. $$

Any ideas or references will be very helpful.

Thank you.

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Maybe the explicit statement of the function $J$ could be helpful? –  awllower May 27 '12 at 7:54
@awllower: Which ine would you propose?I've tried few representations-did not work. Thank you. –  David May 27 '12 at 13:25

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