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 weightwpJ(x,t)=e-txx alpha(1-x)beta,t >= 0,alpha>0,beta>0,x is an element of[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 sigma=2n2t ->infinity,n ->infinity, and Bessel function of order alpha as sigma=2n2t -> 0,n ->infinity; in the neighborhood of x=1, the uniform asymptotic expansion is associated with Bessel function of order beta as n ->infinity. The recurrence coefficients and leading coefficient of the orthogonal polynomials are expressed in terms of a particular Painleve 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 sigma-form Painleve III equation. The asymptotic analysis is based on the Deift and Zhou's steepest descent method.