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

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In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight wpJ(x,t)=etxxα(1x)β,t0,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=2n2t,n, and Bessel function of order α as ς=2n2t0,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.
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Keywords: asymptotic analysis; mathematical physics

Document Type: Research Article

Publication date: July 1, 2019

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