# 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 $\begin{array}{ccc}\hfill \phantom{\rule{40.0pt}{0ex}}& & {w}_{{p}_{J}}\left(x,t\right)={e}^{-\frac{t}{x}}{x}^{\alpha }{\left(1-x\right)}^{\beta },\phantom{\rule{1em}{0ex}}t\ge 0,\hfill \\ \hfill \phantom{\rule{40.0pt}{0ex}}& & \phantom{\rule{1em}{0ex}}\alpha >0,\phantom{\rule{1em}{0ex}}\beta >0,\phantom{\rule{1em}{0ex}}x\in \left[0,1\right].\hfill \end{array}$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 $\varsigma =2{n}^{2}t\to \infty ,n\to \infty$, and Bessel function of order α as $\varsigma =2{n}^{2}t\to 0,n\to \infty ;$ in the neighborhood of $x=1$, the uniform asymptotic expansion is associated with Bessel function of order β as $n\to \infty$. 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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