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 β(t) and the sub-leading coefficient p (n, t) of the monic orthogonal polynomials. This enables us to obtain the large n asymptotics of β(t) and p (n, t) based on the result of Kuijlaars et al. [Adv. Math. 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 β(t). From the t evolution of the auxiliary quantities, we prove that β(t) satisfies a second-order differential equation and R(t) = 2 n+ 1 + 2 α- 2 t(β(t) + β(t)) satisfies a particular Painlevé 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 β(t) and p (n, t).