The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials
The Riemann–Hilbert analysis to the Pollaczek–Jacobi type orthogonal polynomials
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DOI：
10.1111/sapm.12259
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发表时间：
2019-03
影响因子：
2.7
通讯作者：
Min Chen;Yang Chen;E. Fan
中科院分区：
文献类型：
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作者：
Min Chen;Yang Chen;E. Fan
本文研究了关于Pollaczek-Jacobi型权wpJ(x,t)=e−txxα(1−x)β,t≥0,α>,β>0,x∈[0,1]的正交多项式。得到了一元正交多项式在区间（0,1）上和区间外的一致渐近展开式。在x=0附近，一致渐近展开涉及到Airy函数为ς=2n2t→∞,n→∞，α阶Bessel函数为ς=2n2t→0,n→∞；在x=1的邻域中，一致渐近展开与β阶贝塞尔函数n→∞相关联。正交多项式的递归系数和前导系数用一个特定的painlevevleiii超越来表示。我们还得到了谱块核的极限。汉克尔行列式的双尺度对数导数满足σ‐型painleveiii方程。渐近分析基于Deift和Zhou最陡下降法。
In this paper, we study polynomials orthogonal with respect to a Pollaczek–Jacobi type weight wpJ(x,t)=e−txxα(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=2n2t→∞,n→∞ , and Bessel function of order α as ς=2n2t→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.