The Laplace Transform(s) of a Certain Family of Generalized Hypergeometric Functions 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?

Following Wang, I would suggest 
$$\mathcal{L}\{\mathcal{G}_{p}(x);s\}= _{p+1}F_{q}\left[\begin{matrix}
    1,       & a_{1}, & \dots & ,a_{n} \\
           & b_{1}, & \dots & ,b_{q} 
\end{matrix};1/s\right]$$

