A note on the asymptotics of the Hankel determinant associated with time-dependent Jacobi polynomials

Abstract

In 2010, Basor, Chen and Ehrhardt [J. Phys. A 43 (2010), 015204 (25 pp)] studied the (monic) time-dependent Jacobi polynomials. They proved that the diagonal recurrence coefficient α n ( t ) \alpha _n(t) satisfies a particular Painlevé V equation and the sub-leading coefficient p ( n , t ) \mathrm {p}(n,t) satisfies the Jimbo-Miwa-Okamoto σ \sigma -form of Painlevé V under suitable transformations. In the end they made some conjectures about the large n n asymptotics of the associated Hankel determinant. In this note we will derive the large n n asymptotics of the Hankel determinant, together with the recurrence coefficients, the sub-leading coefficient and the square of L 2 L^2 -norm of the time-dependent Jacobi polynomials. We will also show the relation between our problem and Heun’s differential equations.