Raimundas Vidunas, Nico M. Temme
Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function.
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Whittaker functions with one or both parameters large: simplified uniform asymptotic expansions involving Bessel and Airy functions
该研究针对$\kappa \to \infty$时的惠特克函数$W_{\kappa,\mu}(z)$、$M_{\kappa,\mu}(z)$及相关函数，基于关联微分方程推导了涉及贝塞尔函数与艾里函数的简化一致渐近展开式，可覆盖复$z$平面主分支并可扩展至所有分支及负$\mu$。
Simplified Airy function Asymptotic expansions for Reverse Generalised Bessel Polynomials
本文为反向广义贝塞尔多项式推导出基于艾里函数和 Liouville-Green 表示的简化均匀渐近展开，并给出系数函数的显式递推计算，适用于大阶数下包含转向点的广泛参数范围。
Extended Hadamard expansions for the Airy functions
本文提出一种源于最陡下降法且对所有复变量收敛的Airy函数新级数展开，通过基于分支点位置的有限路径分割引入上不完全Gamma函数，并揭示了Stokes现象的路径过渡，仅需少量项即可达到高精度。
We prove the enveloping property of the known divergent asymptotic expansions of the large real zeros of the cylinder and Airy functions, and thereby answering in the affirmative two conjectures posed by Elbert and Laforgia and by Fabijonas and Olver, respectively. The essence of the proof is the construction of analytic functions that return the zeros when evaluated along certain discrete sets of real numbers. By manipulating contour integrals of these functions, we derive the asymptotic expansions of the large zeros truncated after a finite number of terms plus remainders that can be estimat…
Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and slowly varying coefficient functions. The sharpness of the bounds is illustrated numerically with an application to Bessel functions of large order.
Computation of asymptotic expansions of turning point problems via Cauchy's integral formula: Bessel functions
本文提出利用柯西积分公式高精度计算具有大实参数的二阶微分方程转折点问题的渐近展开，重点在于贝塞尔函数的Airy型展开。
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions involve exponential, Airy and Scorer functions and slowly varying analytic coefficient functions involving simple coefficients. The approximations are uniformly valid for large values of the parameter and unbounded real and complex values of the argument. Explicit and readily computable error bounds are either furnished or available for all approximations.
Uniform asymptotic expansions for the zeros of Bessel functions
本研究针对含大参数和简单转向点的常微分方程，提出不含Airy函数导数的重新表述一致渐近展开，进而得到对大阶数所有零点一致有效的贝塞尔函数及其导数零点的显式初等系数渐近展开。
Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(α)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and $α$ small or large, uniformly for unbounded real and complex values of $x$. The new expansions extend the range of computability of $L_n^{(α)}(x)$ compared to previous expansions, in particular with respect to higher terms and large values of $α$. Numerical evidence of their accuracy for real and complex values of $x$ is provided.
Uniform asymptotic expansions are derived for Whittaker's confluent hypergeometric functions $M_{κ,μ}(z)$ and $W_{κ,μ}(z)$, as well as the numerically satisfactory companion function $W_{-κ,μ}(ze^{-πi})$. The expansions are uniformly valid for $μ\rightarrow \infty$, $0 \leq κ/μ\leq 1-δ<1$, and for $0 \leq \arg(z) \leq π$. By using appropriate connection and analytic continuation formulas these expansions can be extended to all unbounded nonzero complex $z$. The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are…
Using a differential equation approach asymptotic expansions are rigorously obtained for Lommel, Weber, Anger-Weber and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The approximations involve Airy and Scorer functions, and are uniformly valid for large real order $ν$ and unbounded complex argument $z$. An interesting complication is the identification of the Lommel functions with the new asymptotic solutions, and in order to do so it is necessary to consider certain sectors of the complex plane, as well as introduce new form…
We construct a new type of convergent asymptotic representations, dyadic factorial expansions. Their convergence is geometric and the region of convergence can include Stokes rays, and often extends down to 0^+. For special functions such as Bessel, Airy, Ei, Erfc, Gamma and others, this region is C without an arbitrarily chosen ray effectively providing uniform convergent asymptotic expansions for special functions. We prove that relatively general functions, Ecalle resurgent ones possess convergent dyadic factorial expansions. We show that dyadic expansions are numerically efficient represen…
General Airy-type equations, heat-type equations and pseudo-processes
该研究系统分析高阶Airy型方程的解、展开式与概率解释，推导相关微分方程，将高阶热型分数阶Cauchy问题基本解用Airy型函数表示，并借助伪过程理论推广结果、研究解的矩。
Uniform asymptotic expansions for the zeros of parabolic cylinder functions
本文针对抛物柱面函数 $U(a,z)$ 的实与复零点，推导了在大 $|a|$ 下对无界 $z$ 一致有效且借助 Airy 函数零点表示的渐近展开，并通过高精度数值对比验证了复零点近似的准确性。
We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of asymptotic analysis will be compared with De Bruijn's book {\em Asymptotic Methods in Analysis}. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are u…
Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros
本文针对虚阶贝塞尔函数，推导了在无界复域中均匀有效且误差界更简单的两类渐近展开，并进一步构造了其零点在大虚阶下无大小限制的均匀渐近展开。
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix theory. Very recently, it has been proved that all rational, symmetric Darboux transformations of the Bessel, Airy, and exponential bispectral functions give rise to commuting integral and differential operators [6, 7, 8], vastly generalizing the known examples in the literature. In this paper, we give a classification of the the rational symmetric Darboux trans…
Airy Turning-Point Asymptotics for Ramanujan's Entire Function $A_q(z)$
本文对拉马努金整函数 $A_q(z)$ 在特定转折点标度下进行渐近分析，得到其艾里函数展开式及一阶修正，确定了其正零点与艾里零点的对应关系，验证了相关差分方程与微分方程的联系，补充了森田反对称伴随函数的归一化极限结果。
The asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$, where $|z|$ may be finite or allowed to be large like $O(n)$, has been recently considered in \cite{P1}. When $z=-x$ is negative, it is found that there is a coalesence of two contributory saddle points when $n/x=1/e$. Here we determine the expansion when $n$ and $x$ satisfy this condition and also a uniform two-term approximation involving the Airy function in the neighbourhood of this value. Numerical results are given to illustrate t…
In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^α e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N} \}$. We focus on the "band-saturated region" situation when the parameter $c > \frac{π^2}{4}$. As $n \to \infty$, uniform expansions for $P_{n, n}(z)$ are achieved for $z$ in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for $z$ near the endpoints of the band and the origin, respectively. The…
In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlevé II differential equation. Using the fact that these tau functions can be written as $n\times n$ Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as $n$-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane.
Numerical methods for the computation of the parabolic cylinder $U(a,z)$ for real $a$ and complex $z$ are presented. The main tools are recent asymptotic expansions involving exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main main methods can be complemented with Maclaurin series and a Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with $5\times 10^{-13}$ relative accuracy in double pr…
Numerical and Asymptotic Aspects of Parabolic Cylinder Functions
本文改进Olver的抛物柱面函数一致渐近展开，用初等函数和Airy函数表示，扩大数值算法适用区间，并给出三组展开的数值测试。
Asymptotic behavior of zeros of Bessel function derivatives
本文导出了贝塞尔函数$n$阶导数零点的两种渐近展开，分别针对$k$或$ν$趋于无穷的情况，并建立显式误差界，扩展并细化了相关经典工作。
Global Rational Approximations of Functions With Factorially Divergent Asymptotic Series
该研究提出新型数值高效的dyadic展开方法，实现阶乘发散渐近级数函数的全局几何收敛有理逼近，适用特殊函数与Écalle再生函数，还可推广至算子预解式表示。
In this paper, we study the strong asymptotic for the orthogonal polynomials and universality associated with singularly perturbed Pollaczek-Jacobi type weight $$w_{p_J2}(x,t)=e^{-\frac{t}{x(1-x)}}x^α(1-x)^β, $$ where $t \ge 0$, $α>0$, $β>0$ and $x \in [0,1].$ Our main results obtained here include two aspects: { I. Strong asymptotics:} We obtain the strong asymptotic expansions for the monic Pollaczek-Jacobi type orthogonal polynomials in different interval $(0,1)$ and outside of interval $\mathbb{C}\backslash (0,1)$, respectively; Due to the effect of $\frac{t}{x(1-x)}$ for varying $t$, diff…
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