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. 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 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 \(\beta _n(t)\) and \(\mathrm {p}(n,t)\).

我们研究由对称半经典雅可比权重生成的正交多项式和 Hankel 行列式。通过使用梯形算子技术，我们推导出由递推系数\(\beta_n(t)\)和次导系数\(\mathrm {p}(n,t)\)满足的二阶非线性差分方程)的单正交多项式。这使我们能够根据 Kuijlaars 等人的结果获得\(\beta _n(t)\)和\(\mathrm {p}(n,t)\)的大n渐近线。[高级。数学。188 (2004) 337–398]。此外，我们展示了正交多项式所满足的二阶微分方程，所有系数都表示为\(\beta _n(t)\)。从辅助量的t演化，我们证明\(\beta _n(t)\)满足二阶微分方程且\(R_n(t)=2n+1+2\alpha -2t(\beta _n (t)+\beta _{n+1}(t))\)在简单变换下满足特定的 Painlevé V 方程。此外，我们表明相关的汉克尔行列式的对数导数满足二阶微分方程和差分方程。Hankel 行列式的大n渐近线是从其积分表示中导出的，以\(\beta _n(t)\)和\(\mathrm {p}(n,t)\) 表示。