  We study orthogonal polynomials and Hankel determinants generated by a
symmetric semi-classical Jacobi weight. By using the ladder operator technique,
we derive the second-order nonlinear difference equations satisfied by the
recurrence coefficient $\beta_n(t)$ and the sub-leading coefficient
$\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to
obtain the large $n$ asymptotics of $\beta_n(t)$ and $\mathrm{p}(n,t)$ based on
the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In
addition, we show the second-order differential equation satisfied by the
orthogonal polynomials, with all the coefficients expressed in terms of
$\beta_n(t)$.
  From the $t$ evolution of the auxiliary quantities, we prove that
$\beta_n(t)$ satisfies a second-order differential equation and
$R_n(t)=2n+1+2\alpha-2t(\beta_n(t)+\beta_{n+1}(t))$ satisfies a particular
Painlev\'{e} V equation under a simple transformation. Furthermore, we show
that the logarithmic derivative of the associated Hankel determinant satisfies
both the second-order differential and difference equations. The large $n$
asymptotics of the Hankel determinant is derived from its integral
representation in terms of $\beta_n(t)$ and $\mathrm{p}(n,t)$.
