Abstract
In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles
here \(\theta (x)\) is the Heaviside function, where \(A> 0\), \(t>0\), and \(z\in [0,\infty )\). We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities \(R_n(t)\) and \(r_n(t)\) satisfy the coupled Riccati equations, from which we also prove that \(R_n(t)\) satisfies a particular Painlevé IV equation. Even more, \(\sigma _n(t)\), allied to \(R_n(t)\), satisfies both the discrete and continuous Jimbo–Miwa–Okamoto \(\sigma \)-form of the Painlevé IV equation. Finally, we consider the situation when n gets large, the second order linear differential equation satisfied by the polynomials \(P_n(x)\) orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.
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Acknowledgements
D. Wang, M. Zhu and Y. Chen would like to give thanks the Science and Technology Development Fund of the Macau SAR for providing FDCT 023/ 2017/A1. They would also like to thank the University of Macau for MYRG 2018-00125 FST. M. Zhu and Y. Chen also acknowledges the support of the Natural Science Foundation of Guangdong Province under Grant No. 2021A1515010361.
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Zhu, M., Wang, D. & Chen, Y. Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities. Anal.Math.Phys. 11, 131 (2021). https://doi.org/10.1007/s13324-021-00560-x
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DOI: https://doi.org/10.1007/s13324-021-00560-x