We study the monic polynomials P(x; t), orthogonal with respect to a symmetric perturbed Gaussian weight function w ( x ) = w ( x ; t ) ≔ e − x 2 1 + t x 2 λ , x ∈ R , with t > 0 , λ ∈ R . This problem is related to single-user multiple-input multiple-output systems in information theory. It is shown that the recurrence coefficient β(t) is related to a particular Painlevé V transcendent, and the sub-leading coefficient p(n, t) of P(x; t) (P(x; t) = x + p(n, t)x + ⋯) satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by β(t) and p(n, t), respectively. This enables us to obtain the large n full asymptotic expansions for β(t) and p(n, t) with the aid of Dyson’s Coulomb fluid approach in the one-cut case [i.e., λt ≤ 1 (t > 0)]. We also consider the Hankel determinant D(t), generated by the perturbed Gaussian weight. It is found that Φ(t), a quantity allied to the logarithmic derivative of D(t) via Φ n ( t ) = 2 t 2 d d t ln D n ( t ) − 2 n λ t , can be expressed in terms of β(t) and p(n, t). Based on this result, we obtain the large n asymptotic expansion of Φ(t) and then that of the Hankel determinant D(t) in the one-cut case.