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The hankel determinants from a singularly perturbed jacobi weight
Han, Pengju1; Chen, Yang2
2021-11-01
Source PublicationMathematics
Volume9Issue:22
AbstractWe study the Hankel determinant generated by a singularly perturbed Jacobi weight (Formula presented). If s = 0, it is reduced to the classical Jacobi weight. For s > 0, the factor (Formula presented) induces an infinitely strong zero at x = 1. For the finite n case, we obtain four auxiliary quantities R (s), r (s), ˜R (s), and ˜r (s) by using the ladder operator approach. We show that the recurrence coefficients are expressed in terms of the four auxiliary quantities with the aid of the compatibility conditions. Furthermore, we derive a shifted Jimbo–Miwa–Okamoto σ-function of a particular Painlevé V for the logarithmic derivative of the Hankel determinant D (s). By variable substitution and some complicated calculations, we show that the quantity R (s) satisfies the four Painlevé equations. For the large n case, we show that, under a double scaling, where n tends to ∞ and s tends to 0, such that τ:= n s is finite, the scaled Hankel determinant can be expressed by a particular P.
KeywordHankel determinant Ladder operators Painlevé V Random matrix theory Singularly perturbed Jacobi weight
DOI10.3390/math9222978
URLView the original
Language英語English
Scopus ID2-s2.0-85119931129
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Document TypeJournal article
CollectionUniversity of Macau
Affiliation1.College of Science, Huazhong Agricultural University, Wuhan, 430070, China
2.Department of Mathematics, Faculty of Science and Technology, University of Macau, 999078, Macao
Recommended Citation
GB/T 7714
Han, Pengju,Chen, Yang. The hankel determinants from a singularly perturbed jacobi weight[J]. Mathematics,2021,9(22).
APA Han, Pengju,&Chen, Yang.(2021).The hankel determinants from a singularly perturbed jacobi weight.Mathematics,9(22).
MLA Han, Pengju,et al."The hankel determinants from a singularly perturbed jacobi weight".Mathematics 9.22(2021).
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