The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials

Abstract

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight truerightleftwpJ(x,t)=e−txxαfalse(1−xfalse)β,1emt≥0,rightleft1emα>0,1emβ>0,1emx∈false[0,1false].The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near x=0, the uniform asymptotic expansion involves Airy function as ς=2n2t→∞,n→∞, and Bessel function of order α as ς=2n2t→0,n→∞; in the neighborhood of x=1, the uniform asymptotic expansion is associated with Bessel function of order β as n→∞. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painlevé III transcendent. We also obtain the limit of the kernel in the bulk of the spectrum. The double scaled logarithmic derivative of the Hankel determinant satisfies a σ‐form Painlevé III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.

Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems

“…Combining (14) with (19), and taking account of (10) and 13, we obtain the expression of ( 1 , 2 ) in terms of the auxiliary quantities…”

“…Comparatively speaking, the RH method is powerful and rigorous in asymptotic analysis of unitary ensembles under double scaling. For problems involving a single variable, the asymptotic expansion of the quantities of interest is characterized by classical Painlevé equations 18–20 . For problems involving ) variables, one can establish a system of coupled ordinary differential equations (ODEs for short) which can be viewed as a generalization of Painlevé equations since the system is reduced to the classical Painlevé equations when is assumed to be 1 or the variables involved take certain values which convert the ‐variable problem to one‐variable problem (see, for instance, Refs.…”

Differential, difference, and asymptotic relations for Pollaczek–Jacobi type orthogonal polynomials and their Hankel determinants

“…The combination of (4), (5), and (6) show that the recurrence coefficient ( ) has two alternative representations:…”

The smallest eigenvalue distribution of the Jacobi unitary ensembles

“…In addition to being applied to derive the constant term in the asymptotics of the gap probability of unitary ensembles, RH method is a powerful tool to study many other problems in large‐dimensional unitary ensembles, for example, the partition function, 14,15 the gap probability distribution, 16,17 the correlation kernel, 18 and orthogonal polynomials 19 . For finite n analysis, the ladder operators adapted to monic orthogonal polynomials are usually used.…”