I am dealing with integrals of the form
$$\int_0^\infty e^{-t}I_0(xt/a)^a\ \mathrm{dt}$$
where $I_0(x)$ is the modified Bessel function of the first kind. Clearly this is just a Laplace Transform $\mathcal{L}(I_0(xt/a)^a)[s]$ in the limit $s\to1$.
A form for similar integrals is given in Prudnikov, Integrals and Series Vol. 4. as below:
\begin{equation} \mathcal{L}\bigg(t^{\lambda}\prod_{k=1}^n I_{v_k}(a_k t)\bigg) = \frac{\Gamma(\lambda+v+1)}{2^vp^{\lambda+v+1}}\bigg[\prod_{k=1}^n \frac{a_k^{v_k}}{\Gamma(v_k+1)}\bigg]F_C^{(n)}\bigg(\lambda+\frac{v}{2}+1,\lambda+\frac{v}{2}+2, v_1+1, ..., v_n+1;\ \frac{a_1^2}{p^2}, ..., \frac{a_n^2}{p^2}\bigg) \end{equation}
Here, $v$ is given as the sum over $k$ of $v_k$ and $F_C$ is a Lauricella hypergeometric series.
This doesn't seem to reproduce correct answers for known results however - e.g, Prudnikov gives
$$\mathcal{L}(x^{1/2}I_{-1/4}(ax)I_{1/4}(ax)) = \sqrt{\frac{2}{\pi p(p^2-4a^2)}}$$
however the formula above evaluates to
$$\sqrt{\frac{2}{\pi p^3}}F_C^{(2)}\bigg(\frac{3}{2},\frac{5}{2},\frac{3}{4},\frac{5}{4}; \frac{a^2}{p^2},\frac{a^2}{p^2}\bigg) = \sqrt{\frac{2}{\pi p^3}}\sum_{n=0}^\infty\sum_{m=0}^\infty \frac{(3/2)_{n+m}(5/2)_{n+m}}{(3/4)_n(5/4)_mn!m!} \bigg(\frac{a^2}{p^2}\bigg)^{n+m}$$
(In the case $n=2$ the Lauricella series corresponds to the Appell $F_4$ series). When evaluating this as a power series it does not align with the power series for the solution given above, and hasn't aligned with other known laplace transforms of products I've tried.
Seemingly the prefactor of $\sqrt{\frac{2}{\pi p^3}}$ is correct as it appears in the series expansion of the square root above.
Is the laplace transform solution as given in Prudnikov correct? I can't find a cross-reference for it or anything else close to it. Any resources or knowledge on the topic would be great.
It appears that this is indeed a slight misprint. Earlier in the book the case
$$ x^\lambda I_\mu(ax)I_\nu(bx) $$ is considered giving rise to
$$ \mathcal{L}(x^\lambda I_\mu(ax)I_\nu(bx))$$ $$= \frac{a^\mu b^\nu}{2^{\mu+\nu}p^{\lambda+\mu+\nu+1}}\Gamma\bigg[{\lambda+\nu+\mu+1 \atop \mu+1,\nu+1}\bigg]F_4\bigg(\frac{\lambda+\mu+\nu+1}{2},\frac{\lambda+\mu+\nu+2}{2};\mu+1,\nu+1;\frac{a^2}{p^2},\frac{b^2}{p^2}\bigg) $$
where $F_4$ is the Appell series. This is suggestive that the first two terms of the Lauricella function given are incorrect, and replacing them by $(\lambda+\mu+\nu+1)/2$ and $(\lambda+\mu+\nu+2)/2$ seems to reproduce known results.