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 weight
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.
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