  We study the monic polynomials orthogonal with respect to a symmetric
perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\:
x^2\right)^\lambda,\qquad x\in \mathbb{R}, $$ where $t> 0,\;\lambda\in
\mathbb{R}$. This weight is related to the single-user MIMO systems in
information theory. It is shown that the recurrence coefficient $\beta_n(t)$ is
related to a particular Painlev\'{e} V transcendent, and the sub-leading
coefficient $\mathrm{p}(n,t)$ satisfies the Jimbo-Miwa-Okamoto $\sigma$-form of
the Painlev\'{e} V equation. Furthermore, we derive the second-order difference
equations satisfied by $\beta_n(t)$ and $\mathrm{p}(n,t)$, respectively. This
enables us to obtain the large $n$ full asymptotic expansions for $\beta_n(t)$
and $\mathrm{p}(n,t)$ with the aid of Dyson's Coulomb fluid approach. We also
consider the Hankel determinant $D_n(t)$, generated by the perturbed Gaussian
weight. It is found that $H_n(t)$, a quantity allied to the logarithmic
derivative of $D_n(t)$, can be expressed in terms of $\beta_n(t)$ and
$\mathrm{p}(n,t)$. Based on this result, we obtain the large $n$ asymptotic
expansion of $H_n(t)$ and then that of the Hankel determinant $D_n(t)$.
