scholarly journals The Hankel Determinants from a Singularly Perturbed Jacobi Weight

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2978
Author(s):  
Pengju Han ◽  
Yang Chen
Keyword(s):  
Logarithmic Derivative ◽  
Singularly Perturbed ◽  
Hankel Determinant ◽  
Compatibility Conditions ◽  
Operator Approach ◽  
Recurrence Coefficients ◽  
Ladder Operator ◽  
Jacobi Weight ◽  
Large N ◽  
Variable Substitution

We study the Hankel determinant generated by a singularly perturbed Jacobi weight w(x,s):=(1−x)α(1+x)βe−s1−x,x∈[−1,1],α>0,β>0s≥0. If s=0, it is reduced to the classical Jacobi weight. For s>0, the factor e−s1−x induces an infinitely strong zero at x=1. For the finite n case, we obtain four auxiliary quantities Rn(s), rn(s), R˜n(s), and r˜n(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 Dn(s). By variable substitution and some complicated calculations, we show that the quantity Rn(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:=n2s is finite, the scaled Hankel determinant can be expressed by a particular PIII′.

The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant

2017 ◽  
Vol 06 (04) ◽  
pp. 1740002 ◽  
Author(s):  
Pengju Han ◽  
Yang Chen
Keyword(s):  
Logarithmic Derivative ◽  
Laguerre Polynomials ◽  
Large Degree ◽  
Hankel Determinant ◽  
Order Ordinary Differential Equation ◽  
Recurrence Coefficients ◽  
Gaussian Unitary Ensemble ◽  
Ladder Operator ◽  
Large N ◽  
Unitary Ensemble

In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight [Formula: see text] Here [Formula: see text], [Formula: see text] and [Formula: see text]. We will describe this problem in terms of the ratio [Formula: see text] where ultimately [Formula: see text] is bounded away from 0, and close to 1. From the ladder operator approach, and the associated compatibility conditions, the recurrence coefficients satisfy a second order ordinary differential equation (ODE) when viewed as functions of the parameter [Formula: see text] in the weight. Then we work out the large degree asymptotics of their recurrence coefficients. We also discuss the associated Hankel determinant. We show that the logarithmic derivative of [Formula: see text] can be expressed in terms of the recurrence coefficients, and obtained the large degree asymptotics of [Formula: see text]. Based on this result, we compute the probability that an [Formula: see text] (or [Formula: see text]) random matrix from a generalized Gaussian Unitary Ensemble (gGUE) is positive definite.

A degenerate Gaussian weight connected with Painlevé equations and Heun equations

2020 ◽  
pp. 2150034
Author(s):  
Pengju Han ◽  
Yang Chen
Keyword(s):  
Logarithmic Derivative ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Operator Technique ◽  
Hankel Determinant ◽  
Recurrence Coefficients ◽  
Hard Edge ◽  
Ladder Operator ◽  
Painlevé Ii ◽  
Gaussian Weight

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight [Formula: see text] where [Formula: see text] and [Formula: see text]. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto [Formula: see text] form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the [Formula: see text]-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case [Formula: see text], we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight [Formula: see text], which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].

Exceptional solutions to the Painlevé VI equation associated with the generalized Jacobi weight

2017 ◽  
Vol 06 (01) ◽  
pp. 1750003
Author(s):  
Shulin Lyu ◽  
Yang Chen
Keyword(s):  
Asymptotic Expansions ◽  
Hankel Determinant ◽  
Generalized Jacobi Weight ◽  
Jacobi Weight ◽  
Special Cases ◽  
Painlevé Vi Equation

We consider the generalized Jacobi weight [Formula: see text], [Formula: see text]. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the [Formula: see text]-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near [Formula: see text] and [Formula: see text] for generic [Formula: see text]. For four special cases of [Formula: see text] which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694 ], and thus give another characterization of the Hankel determinant.

scholarly journals Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble

2019 ◽  
Vol 09 (02) ◽  
pp. 2050016 ◽  
Author(s):  
Chao Min ◽  
Yang Chen
Keyword(s):  
Asymptotic Behavior ◽  
Compatibility Condition ◽  
Logarithmic Derivative ◽  
Hankel Determinant ◽  
Ladder Operators ◽  
Coupled Riccati Equations ◽  
Coulomb Fluid ◽  
Laguerre Unitary Ensemble ◽  
Soft Edge ◽  
Unitary Ensemble

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities [Formula: see text] and [Formula: see text] and show that they satisfy the coupled Riccati equations, from which we find that [Formula: see text] satisfies the Painlevé V equation. Furthermore, we prove that [Formula: see text], a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo–Miwa–Okamoto [Formula: see text]-form of the Painlevé V. In the end, by using Dyson’s Coulomb fluid approach, we consider the large [Formula: see text] asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

scholarly journals On properties of a deformed Freud weight

2018 ◽  
Vol 08 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Mengkun Zhu ◽  
Yang Chen
Keyword(s):  
Hankel Matrix ◽  
Nonlinear Ordinary Differential Equation ◽  
Second Order ◽  
Heun Equation ◽  
Recurrence Coefficients ◽  
Freud Weight ◽  
Ladder Operator ◽  
Linear Ode ◽  
Biconfluent Heun Equation ◽  
Discrete Painlevé Equation

We study the recurrence coefficients of the monic polynomials [Formula: see text] orthogonal with respect to the deformed (also called semi-classical) Freud weight [Formula: see text] with parameters [Formula: see text]. We show that the recurrence coefficients [Formula: see text] satisfy the first discrete Painlevé equation (denoted by d[Formula: see text]), a differential–difference equation and a second-order nonlinear ordinary differential equation (ODE) in [Formula: see text]. Here [Formula: see text] is the order of the Hankel matrix generated by [Formula: see text]. We describe the asymptotic behavior of the recurrence coefficients in three situations: (i) [Formula: see text], [Formula: see text] finite, (ii) [Formula: see text], [Formula: see text] finite, (iii) [Formula: see text], such that the radio [Formula: see text] is bounded away from [Formula: see text] and closed to [Formula: see text]. We also investigate the existence and uniqueness for the positive solutions of the d[Formula: see text]. Furthermore, we derive, using the ladder operator approach, a second-order linear ODE satisfied by the polynomials [Formula: see text]. It is found as [Formula: see text], the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, [Formula: see text], associated with [Formula: see text] when [Formula: see text] tends to infinity.

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