Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics

TL;DR

This paper investigates the asymptotic behavior of orthogonal polynomials, recurrence coefficients, and Hankel determinants generated by a semi-classical Jacobi weight, revealing connections to Painlevé equations and providing detailed asymptotic formulas.

Contribution

It derives nonlinear difference and differential equations for recurrence coefficients and orthogonal polynomials, linking them to Painlevé V and providing large n asymptotics for Hankel determinants.

Findings

Recurrence coefficient β_n(t) satisfies a second-order differential equation.

Hankel determinant's logarithmic derivative obeys differential and difference equations.

Large n asymptotics of Hankel determinants are obtained from integral representations.

Abstract

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