一类广义的十次Freud-型权函数

Abstract

Abstract:

In this paper, the authors focus on a generalised decic Freud-type weight function $w(x;t, \alpha)=|x|^{\alpha}{\rm e}^{-x^{10}+tx^2}, t, x\in\mathbb{R}, \alpha>-1, $ and study the properties of the orthogonal polynomials with respect to this weight. The difference-differential equations of their associated recurrence coefficients are derived; meanwhile, the authors also find the asymptotic behavior of recurrence coefficients via above mentioned equations. What's more, the authors discuss the Hankel determinant in regard to this weight as $n\rightarrow\infty$, and calculate the smallest eigenvalues of large Hankel matrices generated by this weight when $\alpha=t=0$.

Key words: Orthogonal Polynomials, Recurrence coefficients, Hankel determinant, Smallest eigenvalue

CLC Number:

O175.7

Dan Wang,Mengkun Zhu,Chen Yang,Xiaoli Wang. A Generalised Decic Freud-Type Weight[J].Acta mathematica scientia,Series A, 2021, 41(4): 921-935.

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Figures/Tables 5

References 38

1	Chihara T S . An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978
2	Freud G . Othogonal Polynomials. Budapest, Hungary: Pergamon, 1971
3	Van Assche W . Asymptotics for Orthogonal Polynomials. Berlin: Springer, 1987
4	Freud G . On the coefficients in the recursion formulae of orthogonal polynomials. Proc R Irish Acad, 1976, 76, 1- 6
5	Magnus A P. A Proof of Freud's Conjecture About the Orthogonal Polynomials Related to $\|x\|^{\rho}\exp(-x^{2m})$, for Integer $m$//Brezinski C, Draux A, Magnus A P, et al. Orthogoanl Polynomials and Applications. Berlin: Springer-Verlag, 1985: 362-372
6	Lubinsky D , Mhaskar H , Saff E . A proof of Freud's conjecture for exponential weights. Constr Approx, 1988, 4, 65- 83 doi: 10.1007/BF02075448
7	Magnus A P . On Freud's equations for the exponential weights. J Approx Theory, 1986, 46, 65- 99 doi: 10.1016/0021-9045(86)90088-2
8	Clarkson P A , Jordaan K , Kelil A . A generalized Freud weight. Stud Appl Math, 2016, 136, 288- 320 doi: 10.1111/sapm.12105
9	Clarkson P A , Jordaan K . Properties of generalized Freud polynomials. J Approx Theory, 2018, 25, 148- 175
10	Bleher P , Its A R . Semiclassical asymptotics of orthogonal polynomials: Riemann-Hilbert problem, and universality in the matrix model. Ann of Math, 1999, 150, 185- 266 doi: 10.2307/121101
11	Bleher P , Its A R . Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm Pure Appl Math, 2003, 56, 433- 516 doi: 10.1002/cpa.10065
12	Clarkson P A , Jordaan K . The relationship between semiclassical Laguerre polynomials and the fourth Painlevé equation. Constr Approx, 2014, 39, 223- 254 doi: 10.1007/s00365-013-9220-4
13	Levin E , Lubinsky D . Orthogonal Polynomials with Exponential Weights. New York: Springer-Verlag, 2001
14	Lyu S L , Chen Y . The largest eigenvalue distribution of the Laguerre unitary ensemble. Acta Math Scientia, 2017, 37 (2): 439- 462 doi: 10.1016/S0252-9602(17)30013-9
15	Wong R , Zhang L . Global asymptotics of orthogonal polynomials associated with $\|x\|^{2\alpha}{\rm e}^{-Q(x)}$. J Approx Theory, 2010, 162, 723- 765 doi: 10.1016/j.jat.2009.09.007
16	Bonan S , Clark D S . Estimates of the orthogonal polynomials with weight $\exp(-x^m)$, $m$ an even positive integer. J Approx Theory, 1986, 46, 408- 410 doi: 10.1016/0021-9045(86)90074-2
17	Bonan S , Nevai P . Orthogonal polynomials and their derivatives I. J Approx Theory, 1984, 40, 134- 147 doi: 10.1016/0021-9045(84)90023-6
18	Chen Y , Pruessner G . Orthogonal polynomials with discontinuous weights. J Phys A: Math Gen, 2005, 38, 191- 198 doi: 10.1088/0305-4470/38/12/L01
19	Wang D , Zhu M K , Chen Y . On semi-classical orthogonal polynomials associated with a Freud - type weight. Math Meth Appl Sci, 2020, 43 (8): 5295- 5313 doi: 10.1002/mma.6270
20	Vein P R , Dale P . Determinants and Their Applications in Mathematical Physics. New York: Springer-Verlag, 1999
21	Van Assche W. Discrete Painlevé Equations for Recurrence Coefficients of Orthogonal Polynomials//Elaydi S, Cushing J, Lasser R, et al. Difference Equations, Special Functions and Orthogonal Polynomials. Hackensack: World Scientific, 2007: 687-725
22	Cresswell C , Joshi N . The discrete first, second and thirty-fourth Painlevé hierarchies. J Phys A: Math Gen, 1999, 32, 655- 669 doi: 10.1088/0305-4470/32/4/009
23	Damelin S B . Asymptotics of recurrence coefficients for orthonormal polynomials on the line-Magnus method revisited. Math Comp, 2004, 73, 191- 209
24	Chen Y , Ismail M E H . Thermodynamic relations of the Hermitian matrix ensembles. J Phys A: Math Gen, 1997, 30, 6633- 6654 doi: 10.1088/0305-4470/30/19/006
25	Chen Y , Lawrence N . On the linear statistics of Hermitian random matrices. J Phys A: Math Gen, 1998, 31, 1141- 1152 doi: 10.1088/0305-4470/31/4/005
26	Chen Y , Mckay M R . Coulomb fluid, Painlevé transcendents and the information theory of MIMO systems. IEEE Trans Inform Theory, 2012, 58, 4594- 4634 doi: 10.1109/TIT.2012.2195154
27	Zhu M K , Chen Y . On properties of a deformed Freud weight. Random Matrices Theory Appl, 2019, 8, 1950004 doi: 10.1142/S2010326319500047
28	Berg C , Chen Y , Ismail M E H . Small eigenvalues of large Hankel matrices: The indeterminate case. Math Scand, 2002, 91, 67- 81 doi: 10.7146/math.scand.a-14379
29	Berg C , Szwarc R . The smallest eigenvalue of Hankel matrices. Const Approx, 2011, 34, 107- 133 doi: 10.1007/s00365-010-9109-4
30	Chen Y , Lawrence N D . Small eigenvalues of large Hankel matrices. J Phys A: Math Gen, 1999, 32, 7305- 7315 doi: 10.1088/0305-4470/32/42/306
31	Chen Y , Lubinsky D S . Smallest eigenvalues of Hankel matrices for exponential weights. J Math Anal Appl, 2004, 293, 476- 495 doi: 10.1016/j.jmaa.2004.01.032
32	Emmart N , Chen Y , Weems C . Computing the smallest eigenvalue of large ill-conditioned Hankel matrices. Commun Comput Phys, 2015, 18, 104- 124 doi: 10.4208/cicp.260514.231214a
33	Szegö G . On some Hermitian forms associated with two given curves of the complex plane. Trans Amer Math Soc, 1936, 40, 450- 461 doi: 10.1090/S0002-9947-1936-1501884-1
34	Widom H , Wilf H S . Small eigenvalues of large Hankel matrices. Proc Amer Math Soc, 1966, 17, 338- 344 doi: 10.1090/S0002-9939-1966-0189237-7
35	Zhu M K , Chen Y , Emmart N , Weems C . The smallest eigenvalue of large Hankel matrices. Appl Math Comput, 2018, 334, 375- 387
36	Zhu M K , Emmart N , Chen Y , Weems C . The smallest eigenvalue of large Hankel matrices generated by a deformed Laguerre weight. Math Meth Appl Sci, 2019, 42, 3272- 3288 doi: 10.1002/mma.5583
37	Chen Y , Sikorowski J , Zhu M K . Smallest eigenvalue of large Hankel matrices at critical point: Comparing conjecture with parallelised computation. Appl Math Comput, 2019, 363, 124628- 124646
38	Zhu M K , Chen Y , Li C Z . The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight. J Math Phys, 2020, 61, 073502 doi: 10.1063/1.5140079

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A Generalised Decic Freud-Type Weight

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