scholarly journals Painlevé VI, Painlevé III, and the Hankel determinant associated with a degenerate Jacobi unitary ensemble

2020 ◽  
Vol 43 (15) ◽  
pp. 9169-9184
Author(s):  
Chao Min ◽  
Yang Chen
Keyword(s):  
Hankel Determinant ◽  
Unitary Ensemble

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.

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 Coefficients problems for families of holomorphic functions related to hyperbola

2020 ◽  
Vol 70 (3) ◽  
pp. 605-616
Author(s):  
Stanisława Kanas ◽  
Vali Soltani Masih ◽  
Ali Ebadian
Keyword(s):  
Unit Disk ◽  
Holomorphic Functions ◽  
Hankel Determinant ◽  
Coefficient Problem

AbstractWe consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

scholarly journals Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Massimo Gisonni ◽  
Tamara Grava ◽  
Giulio Ruzza
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Hurwitz Numbers ◽  
Jacobi Ensemble ◽  
Matrix Ensembles ◽  
Wilson Polynomials ◽  
Unitary Ensemble ◽  
Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

scholarly journals On Coefficient Problems for Functions Connected with the Sine Function

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1179
Author(s):  
Katarzyna Tra̧bka-Wiȩcław
Keyword(s):  
Analytic Functions ◽  
Sine Function ◽  
Hankel Determinant ◽  
Coefficient Problems ◽  
Logarithmic Coefficients ◽  
Symmetric Points

In this paper, some coefficient problems for starlike analytic functions with respect to symmetric points are considered. Bounds of several coefficient functionals for functions belonging to this class are provided. The main aim of this paper is to find estimates for the following: coefficients, logarithmic coefficients, some cases of the generalized Zalcman coefficient functional, and some cases of the Hankel determinant.

scholarly journals The second Hankel determinant for strongly convex and Ozaki close-to-convex functions

2021 ◽  
Author(s):  
Young Jae Sim ◽  
Adam Lecko ◽  
Derek K. Thomas
Keyword(s):  
Unit Disk ◽  
Convex Functions ◽  
Univalent Functions ◽  
Inverse Function ◽  
Invariance Property ◽  
Hankel Determinant ◽  
Sharp Bounds ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
Second Hankel Determinant

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${{\mathcal {S}}}$$ S be the subclass of normalized univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . We give sharp bounds for the modulus of the second Hankel determinant $$ H_2(2)(f)=a_2a_4-a_3^2$$ H 2 ( 2 ) ( f ) = a 2 a 4 - a 3 2 for the subclass $$ {\mathcal F_{O}}(\lambda ,\beta )$$ F O ( λ , β ) of strongly Ozaki close-to-convex functions, where $$1/2\le \lambda \le 1$$ 1 / 2 ≤ λ ≤ 1 , and $$0<\beta \le 1$$ 0 < β ≤ 1 . Sharp bounds are also given for $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | , where $$f^{-1}$$ f - 1 is the inverse function of f. The results settle an invariance property of $$|H_2(2)(f)|$$ | H 2 ( 2 ) ( f ) | and $$|H_2(2)(f^{-1})|$$ | H 2 ( 2 ) ( f - 1 ) | for strongly convex functions.

Third Hankel determinants for two classes of analytic functions with real coefficients

2021 ◽  
Vol 33 (4) ◽  
pp. 973-986
Author(s):  
Young Jae Sim ◽  
Paweł Zaprawa
Keyword(s):  
Analytic Functions ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Hankel Determinant ◽  
Lower And Upper Bounds ◽  
Hankel Determinants ◽  
The Third ◽  
Typically Real ◽  
Class Of Functions ◽  
Typically Real Functions

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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