  In this paper, we study the orthogonal polynomials with respect to a
singularly perturbed Pollaczek-Jacobi type weight $$
w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad
x\in[-1,1],\;\;\alpha>0,\;\;t>0. $$ By using the ladder operator approach, we
establish the second-order difference equations satisfied by the recurrence
coefficient $\beta_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of
the monic orthogonal polynomials, respectively. We show that the logarithmic
derivative of $\beta_n(t)$ can be expressed in terms of a particular
Painlev\'{e} V transcendent. The large $n$ asymptotic expansions of
$\beta_n(t)$ and $\mathrm{p}(n,t)$ are obtained by using Dyson's Coulomb fluid
method together with the related difference equations.
  Furthermore, we study the associated Hankel determinant $D_n(t)$ and show
that a quantity $\sigma_n(t)$, allied to the logarithmic derivative of
$D_n(t)$, can be expressed in terms of the $\sigma$-function of a particular
Painlev\'{e} V. The second-order differential and difference equations for
$\sigma_n(t)$ are also obtained. In the end, we derive the large $n$
asymptotics of $\sigma_n(t)$ and $D_n(t)$ from their relations with
$\beta_n(t)$ and $\mathrm{p}(n,t)$.
