Orthogonal polynomials, asymptotics, and Heun equations

Chen, Yang; Filipuk, Galina; Zhan, Longjun

doi:10.1063/1.5102102

Cited by 14 publications

(2 citation statements)

References 47 publications

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“…one would find that this is the biconfluent Heun equation (BHE) [23, p. 194 (1.2.5)]. The relations between orthogonal polynomials and Heun's differential equations have been discussed in recent years; see [24][25][26] for reference.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Painlevé IV, Chazy II, and asymptotics for recurrence coefficients of semi‐classical Laguerre polynomials and their Hankel determinants

Min

Chen

2023

Math Methods in App Sciences

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This paper studies the monic semi‐classical Laguerre polynomials based on previous work by Boelen and Van Assche, Filipuk et al., and Clarkson and Jordaan. Filipuk et al. proved that the diagonal recurrence coefficient satisfies the fourth Painlevé equation. In this paper, we show that the off‐diagonal recurrence coefficient fulfills the first member of Chazy II system. We also prove that the sub‐leading coefficient of the monic semi‐classical Laguerre polynomials satisfies both the continuous and discrete Jimbo–Miwa–Okamoto ‐form of Painlevé IV. By using Dyson's Coulomb fluid approach together with the discrete system for and , we obtain the large asymptotic expansions of the recurrence coefficients and the sub‐leading coefficient. The large asymptotics of the associated Hankel determinant (including the constant term) is derived from its integral representation in terms of the sub‐leading coefficient.

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Section: Statement Of Main Resultsmentioning

confidence: 99%

Painlevé IV, Chazy II, and asymptotics for recurrence coefficients of semi‐classical Laguerre polynomials and their Hankel determinants

Min

Chen

2023

Math Methods in App Sciences

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“…When γ = 0, (6.10) satisfies the bi-confluent Heun equation. More details of the Heun equation, see[5,10,16,18,19,21]. .…”

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Painlevé IV, $σ-$Form and the Deformed Hermite Unitary Ensembles

Zhu,

Wang,

Chen

2021

Preprint

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We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z, t, γ) = e −z 2 +tz |z − t| γ (A + Bθ(z − t)), where A ≥ 0, A + B ≥ 0, t ∈ R, γ > −1 and z ∈ R. By using the ladder operators for the corresponding monic orthogonal polynomials, and their relative compatibility conditions, we obtain a series of difference and differential equations to describe the relations among α n , β n , R n (t) and r n (t). Especially, we find that the auxiliary quantities R n (t) and r n (t) satisfy the coupled Riccati equations, and R n (t) satisfies a particular Painlevé IV equation. Based on above results, we show that σ n (t) and σn (t), two quantities related to the Hankel determinant and R n (t), satisfy the continuous and discrete σ−form equations, respectively. In the end, we also discuss the large n asymptotic behavior of R n (t), which produce the expansion of the logarithmic of the Hankel determinant and the asymptotic of the second order differential equation of the monic orthogonal polynomials.

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Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices

Wang,

Zhu

2024

Math Phys Anal Geom

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Orthogonal polynomials, asymptotics, and Heun equations

Cited by 14 publications

References 47 publications

Painlevé IV, Chazy II, and asymptotics for recurrence coefficients of semi‐classical Laguerre polynomials and their Hankel determinants

Painlevé IV, Chazy II, and asymptotics for recurrence coefficients of semi‐classical Laguerre polynomials and their Hankel determinants

Painlevé IV, $σ-$Form and the Deformed Hermite Unitary Ensembles

Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices

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