I am interested in any analytic information about the following integral:
$i^{4m+1} \int_0^{\infty} t^{1/4} J_m^4(t) J_{\nu}(\alpha t) dt$
where
- $i = \sqrt{-1}$ is the imaginary unit
- $m$ is a non-negative integer
- $J_m(t)$ is the Bessel function of the first kind
- $\nu = \frac{1}{2}$ or $\nu = -\frac{1}{2}$
- $\alpha > 0$
I have looked through some of the common resources (Abramowitz and Stegun, Gradshteyn and Ryzhik, Watson's treatise) and haven't seen any results that seem immediately applicable. If anyone has any further resources or useful identities I am all ears!
Considering (for the given conditions $m\geq 0$ and $\alpha >0$) $$I_\nu=\int_0^{\infty} t^{1/4} J_m^4(t) J_{\nu}(\alpha t) dt$$ and using a CAS
$$I_{+\frac12}=\frac{ \sin \left(\frac{4m+3}{8} \pi \right)\, \Gamma \left(\frac{4m+3}{4}\right) }{2^{m-\frac{1}{2}}\,\alpha ^{m+\frac{5}{4}}\,\sqrt{\pi }}\,\,\, _2\tilde{F}_1\left(\frac{4m+3}{8} ,\frac{4m+7}{8} ;m+1;\frac{1}{\alpha ^2}\right)$$
$$I_{-\frac12}=\frac{\cos \left(\frac{4m+3}{8} \pi \right)\, \Gamma \left(\frac{4m+3}{4}\right) }{2^{m-\frac{1}{2}}\, \alpha ^{m+\frac{5}{4}}\, \sqrt{\pi }}\,\,\, _2\tilde{F}_1\left(\frac{4m+3}{8} ,\frac{4m+7}{8} ;m+1;\frac{1}{\alpha ^2}\right)$$ where appears the regularized hypergeometric function.