arXiv subjects

Yu-Tian Li

Publications and source records attributed to Yu-Tian Li.

9 recordsLinked to original sources

arXiv2026

The Erd\'elyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials

Let $p_n^{(\alpha,\beta)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^\alpha(1+x)^\beta$ on $[-1,1]$, where $\alpha,\beta\ge-1/2$, and put $S=\alpha+\beta+1$. We prove the uniform degree--parameter estimate $$ (1-x)^{\alpha+1/2}(1+x)^{\beta+1/2} \bigl|p_n^{(\alpha,\beta)}(x)\bigr|^2 \le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\bigr\}. $$ This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erd\'elyi--Magnus--Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor $S^{1/3}$ in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss--Jacobi quadrature weights.

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arXiv2026

The Koornwinder--Kostenko--Teschl Conjecture for Jacobi Polynomials and the Discrete Laguerre Phase Transition

We prove the refined Koornwinder--Kostenko--Teschl conjecture. For the normalized weighted Jacobi function and all $n\in\mathbb N\_0$, $\alpha,\beta\ge0$, and $-1\le x\le1$, $$ |g\_n^{(\alpha,\beta)}(x)| \le \left[ \frac{(n+1)(n+\alpha+\beta+1)} {(n+\alpha+1)(n+\beta+1)} \right]^{1/4}\le1. $$ The proof combines a central contour estimate with Sturm--Sonin localization and an exact inverse moment. It also yields an $n$-uniform extreme-lobe theorem and a sharp canonical-product first-lobe principle. Applied to the discrete Laguerre evolution, the estimate gives the optimal positive-parameter decay. For $-1<\alpha\le0$, a complementary one-sided Jacobi inequality gives the exact norm $\|\mathrm{e}^{-\mathrm{i} tH_\alpha}\|_{\ell^1\to\ell^\infty} =(1+t^2)^{-(1+\alpha)/2}$. Thus the large-time decay exponent is $\min\{1,1+\alpha\}$ for every $\alpha>-1$. Bessel, Laguerre, and Darboux scaling limits show that the temporal and fixed-diagonal exponents are optimal.

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arXiv2026

Airy Turning-Point Asymptotics for Ramanujan's Entire Function $A_q(z)$

Let $$ A_q(z)=\sum_{k=0}^{\infty}\frac{q^{k^2}}{(q;q)_k}(-z)^k $$ be Ramanujan's entire function. We study it in the turning-point scaling $$ q=e^{-\varepsilon},\qquad z=\frac{\sqrt q}{4}e^{-\varepsilon^{2/3}\zeta}, $$ with $\zeta$ in a compact subset of $\mathbb{C}$. After an explicit exponential normalization, a direct coalescing-saddle analysis gives, uniformly on compact subsets, the expansion $$ \operatorname{Ai}(\zeta) +\frac{\varepsilon^{2/3}}{30} \left(4\zeta\operatorname{Ai}(\zeta) +\zeta^2\operatorname{Ai}'(\zeta)\right) +O_K(\varepsilon). $$ Thus the first correction is explicit and comes with a quantitative remainder. For every fixed $n$, the same analysis locates the positive zero associated with the $n$-th Airy zero and proves that it is globally the $n$-th positive zero of $A_q$. Expanding the normalized $q$-difference equation recovers the Airy differential equation and confirms the scaling. An appendix records Morita's antisymmetric companion, whose normalized limit is $\operatorname{Bi}$, together with a single-valued meromorphic descent of it. Numerical tables illustrate the normalization, the correction, and the zero formulas.

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arXiv2025

Classification of the real Painlevé I transcendents by zeros and connection problem: an asymptotic study

In this paper, we study the asymptotic behavior and connection problem of Painlevé I (PI) equation through a detailed analysis of the Stokes multipliers associated with its solutions. Focusing on the regime where the derivative at the real zeros of the solution becomes large, we apply the complex WKB method to derive full asymptotic expansions of the Stokes multipliers. These expansions allow us to classify real solutions of PI according to their behavior at the zeros, distinguishing between oscillatory, separatrix, and singular types solutions on the negative real axis. Furthermore, we resolve the connection problem between the large negative asymptotics and the location of positive zeros by establishing full asymptotic expansions of the zero parameters. Our approach enables the construction of a precise phase diagram in the $(r,b)$-plane, where $r$ is the location of a zero and $b$ is the derivative at that point. Numerical simulations are provided to validate the theoretical results. This work extends prior studies on monodromy asymptotics and contributes a comprehensive framework for understanding the global structure of real PI solutions through their local zero data.

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arXiv2025

Full Asymptotic Expansion of Monodromy Data for the First Painlevé Transcendent: Applications to Connection Problems

We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlevé transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex WKB method, also known as the method of uniform asymptotics, to approximate the second-order ODEs derived from PI's Lax pair with higher-order accuracy. As an application, we provide a rigorous proof of the full asymptotic expansion of the nonlinear eigenvalues proposed numerically by Bender, Komijani, and Wang. Additionally, we present the full asymptotic expansion for the pole parameters $(p_{n}, H_{n})$ corresponding to the $n$-th pole of the real tritronquée solution of the PI equation as $n \to +\infty$.

nlin.SI↗

arXiv2023

Connection problem of the first Painlevé transcendents with large initial data

In previous work, Bender and Komijani (2015 \textit{J. Phys. A: Math. Theor.} 48, 475202) studied the first Painlevé (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a $\mathcal{PT}$-symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions $(y(0),y'(0))=(a,b)$ located on a sequence of curves $Γ_n$, $n=1,2,\dots$, will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting form equation $b^2/4 - a^3=f_n \sim A n^{6/5}$ for the curves $Γ_{n}$ as $n\to\infty$ is derived, where $A$ is a positive constant. The discrete set $\{f_n\}$ could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of $Γ_n$ matches the numerical results remarkably well, even for small $n$. The main tool is the method of uniform asymptotics introduced by Bassom et al. (1998 \textit{Arch. Rational Mech. Anal.} {143}, 241--271) in the studies of the second Painlevé equation.

nlin.SI↗

arXiv2023

Connection problem of the first Painlevé transcendent between poles and negative infinity

We consider a connection problem of the first Painlevé equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real $\mathrm{P_I}$ functions. We get a classification of the real $\mathrm{P_I}$ functions in terms of $(p,H)$ so that they behave differently at the negative infinity, where $p$ is the location of a pole and $H$ is the free parameter in the Laurent series. Some limiting-form connection formulas of $\mathrm{P_I}$ functions are obtained for large $H$. Specifically, for the real tritronquée solution, the large-$n$ asymptotic formulas of $p_n$ and $H_n$ are obtained, where $p_n$ is the $n$-th pole on the real line in the ascending order and $H_n$ is the associated free parameter. Our approach is based on the complex WKB method (also known as the method of uniform asymptotics) introduced by Bassom, Clarkson, Law and McLeod in their study on the connection problem of the second Painlevé transcendent [Arch. Rational Mech. Anal., 1998, pp. 241-271]. Several numerical simulations are carried out to verify our main results. Meanwhile, we obtain the phase diagram of \PI~solutions in the $(p,H)$ plane, which somewhat resembles the Brillouin zones in solid-state physics. The asymptotic and numerical results obtained in this paper partially answer Clarkson's open question on the connection problem of the first Painlevé transcendent.

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arXiv2019

Asymptotic approximations of the continuous Hahn polynomials and their zeros

Asymptotic approximations for the continuous Hahn polynomials and their zeros as the degree grows to infinity are established via their three-term recurrence relation. The methods are based on the uniform asymptotic expansions for difference equations developed by Wang and Wong (\textit{Numer. Math.}, 2003) and the matching technique in the complex plane developed by Wang (\textit{J. Approx. Theory}, 2014).

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arXiv2017

Real solutions of the first Painlevé equation with large initial data

We consider three special cases of the initial value problem of the first Painlevé equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of the PI solutions on the negative real axis, recently revealed by Bender and Komijani, is given by approximating the Stokes multipliers. Moreover, we build more precise relation between the large initial data of the PI solutions and their three different types of behavior as the independent variable tends to negative infinity. In addition, some limiting form connection formulas are obtained.

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