Using the standard notation for a generalized hypergeometric function, given a non-negative integer $p$, define:
$\mathcal{G}_{p}\left(x\right)={}_{p}F_{p}\left(\underbrace{\frac{1}{2},...,\frac{1}{2}}_{p\textrm{ times}};\underbrace{\frac{3}{2},...,\frac{3}{2}}_{p\textrm{ times}};-x^{2}\right)$
for all $x$ (in $\mathbb{R}$, or in $\mathbb{C}$). That is to say (as can be easily shown):
$\mathcal{G}_{p}\left(x\right)=3^{p}\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{n!}\frac{x^{2n}}{\left(2n+3\right)^{p}}$
This function is integrable in the nicest possible ways (it is a schwartz function, rapidly decreasing, &c., &c....) and, it is analytic everywhere.
I would like to be able to find the laplace transforms ($\mathcal{L}\left\{ \mathcal{G}_{p}\right\} \left(s\right) )$ of the $\mathcal{G}_{p}$s for every $p$ . At a minimum, I want a closed-form expression for the value of the laplace transforms at $s=1$ ; i.e., the value of the integral:
$\int_{0}^{\infty}\mathcal{G}_{p}\left(x\right)e^{-x}dx$
I cannot do this integration term-by-term, since that leads to a divergent series (and is not valid, due to an absence of uniform convergence of the integrand for $x\in\mathbb{R}\geq0$).
Any ideas?