Dan Dai

Painlevé VI and Hankel determinants for the generalized Jacobi weight

We study the Hankel determinant of the generalized Jacobi weight (x − t)γxα(1 − x)β for x [0, 1] with α, β > 0, t < 0 and . Based on the ladder operators for the corresponding monic orthogonal polynomials Pn(x), it is shown that the logarithmic derivative of the Hankel determinant is characterized by a Jimbo–Miwa–Okamoto σ-form of the Painleve VI system.

Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight

Abstract In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painleve III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method.

Asymptotics for Laguerre polynomials with large order and parameters

We study the asymptotic behavior of Laguerre polynomials L n ( α n ) ( z ) as n ? ∞ , where α n / n has a finite positive limit or the limit is + ∞ . Applying the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we derive the uniform asymptotics of such polynomials, which improves on the results of Bosbach and Gawronski (1998). In particular, our theorem is useful to obtain the asymptotics of complex Hermite polynomials and related double integrals.

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

We study the Fredholm determinant of an integrable operator acting on the interval

Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

(0,s)

Global Asymptotics of Krawtchouk Polynomials -- a Riemann-Hilbert Approach

whose kernel is constructed out of a hierarchy of higher order analogues to the Painleve III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order

Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials

k

Painlevé IV asymptotics for orthogonal polynomials with respect to a modified Laguerre weight

in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large

On tronquée solutions of the first Painlevé hierarchy

s

Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

asymptotics of the Fredholm determinant. Moreover, we derive a Painleve type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to the coupled Painleve III system.