Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics

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 $\mathrm{p}(n,t)$.