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Semi-classical Jacobi polynomials, Hankel determinants and asymptotics

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Abstract

We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient \(\beta _n(t)\) and the sub-leading coefficient \(\mathrm {p}(n,t)\) of the monic orthogonal polynomials. This enables us to obtain the large n asymptotics of \(\beta _n(t)\) and \(\mathrm {p}(n,t)\) based on the result of Kuijlaars et al. [Adv. Math. 188 (2004) 337–398]. In addition, we show the second-order differential equation satisfied by the orthogonal polynomials, with all the coefficients expressed in terms of \(\beta _n(t)\). From the t evolution of the auxiliary quantities, we prove that \(\beta _n(t)\) satisfies a second-order differential equation and \(R_n(t)=2n+1+2\alpha -2t(\beta _n(t)+\beta _{n+1}(t))\) satisfies a particular Painlevé V equation under a simple transformation. Furthermore, we show that the logarithmic derivative of the associated Hankel determinant satisfies both the second-order differential and difference equations. The large n asymptotics of the Hankel determinant is derived from its integral representation in terms of \(\beta _n(t)\) and \(\mathrm {p}(n,t)\).

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Acknowledgements

The authors would like to thank the referee for helpful comments which substantially improved the presentation of this paper. The work of Chao Min was partially supported by the National Natural Science Foundation of China under grant number 12001212, by the Fundamental Research Funds for the Central Universities under grant number ZQN-902 and by the Scientific Research Funds of Huaqiao University under grant number 17BS402. The work of Yang Chen was partially supported by the Macau Science and Technology Development Fund under grant number FDCT 023/2017/A1 and by the University of Macau under grant number MYRG 2018-00125-FST.

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, China

    Chao Min

  2. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau, China

    Yang Chen

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  1. Chao Min
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Correspondence to Chao Min.

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Min, C., Chen, Y. Semi-classical Jacobi polynomials, Hankel determinants and asymptotics. Anal.Math.Phys. 12, 8 (2022). https://doi.org/10.1007/s13324-021-00619-9

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  • DOI: https://doi.org/10.1007/s13324-021-00619-9

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