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

In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weightw p J( x , t ) = e − t xx α( 1 − x ) β , t ≥ 0 ,α > 0 , β > 0 , x ∈ [ 0 , 1 ] .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 = 2 n 2 t → ∞ , n → ∞ , and Bessel function of order α as ς = 2 n 2 t → 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.