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

Abstract

AbstractIn this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight urn:x-wiley:00222526:media:sapm12259:sapm12259-math-0001The uniform asymptotic expansions for the monic orthogonal polynomials on the interval (0,1) and outside this interval are obtained. Moreover, near , the uniform asymptotic expansion involves Airy function as , and Bessel function of order α as in the neighborhood of , the uniform asymptotic expansion is associated with Bessel function of order β as . 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.