Asymptotic expansion of holonomic distributions of one complex variable

On the Asymptotic Expansion of the $q$-Dilogarithm

In this work, we study some asymptotic expansion of the $q$-dilogarithm at $q=1$ and $q=0$ by using Mellin transform and adequate decomposition allowed by Lerch functional equation.

Asymptotic expansions for ratios of products of gamma functions

An asymptotic expansion for a ratio of products of gamma functions is derived.

Parabolic Cylinder Functions: Examples of Error Bounds For Asymptotic Expansions

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular Poincar{é}-type expansions for large values of the argument $z$ and uniform expansions for large values of the parameter are considered. The expansions are based on those given in Olver(1959) and on modifications of these expansions given in Temme(2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations.

广义三角积分及其零点的均匀渐近展开

Uniform asymptotic expansions for generalised trigonometric integrals and their zeros

本文提出广义三角积分及其实零点的均匀渐近展开，适用于大参数和无界复值，基于不完全伽马函数的Liouville-Green展开。

Uniform Asymptotic Expansion for the Incomplete Beta Function

In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the incomplete beta function was derived. It was not obvious from those results that the expansion is actually an asymptotic expansion. We derive a remainder estimate that clearly shows that the result indeed has an asymptotic property, and we also give a recurrence relation for the coefficients.

Asymptotic expansion of Mathieu power series and trigonometric Mathieu series

We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series.

The higher order asymptotic expansion of the Krawtchouk polynomials

The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few initial terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation.

An asymptotic expansion for a ratio of products of gamma functions

An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.

Asymptotic expansions of integral means and applications to the ratio of gamma functions

Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form $B(A(x))=C(x)$, where $B$ and $C$ have known asymptotic expansions. The results are illustrated by calculation of some important integral means connected with gamma and digamma functions.

An asymptotic expansion for the Stieltjes constants

The Stieltjes constants $γ_n$ appear in the coefficients in the Laurent expansion of the Riemann zeta function $ζ(s)$ about the simple pole $s=1$. We present an asymptotic expansion for $γ_n$ as $n\rightarrow \infty$ based on the approach described by Knessel and Coffey [Math. Comput. {\bf 80} (2011) 379--386]. A truncated form of this expansion with explicit coefficients is also given. Numerical results are presented that illustrate the accuracy achievable with our expansion.

Complete asymptotic expansions for the relativistic Fermi-Dirac integral

Fermi-Dirac integrals appear in problems in nuclear astrophysics, solid state physics or in the fundamental theory of semiconductor modeling, among others areas of application. In this paper, we give new and complete asymptotic expansions for the relativistic Fermi-Dirac integral. These expansions could be useful to obtain a correct qualitative understanding of Fermi systems. The performance of the expansions is illustrated with numerical examples.

库仑波函数零点的McMahon型渐近展开

McMahon-type asymptotic expansions of the zeros of the Coulomb wave functions

本文推导了库仑波函数及其导数大零点的McMahon型渐近展开，该展开在参数为零时退化为贝塞尔函数零点的McMahon展开，并通过数值测试验证了其精度。

Asymptotic expansion of the modified exponential integral involving the Mittag-Leffler function

We consider the asymptotic expansion of the generalised exponential integral involving the Mittag-Leffler function introduced recently by Mainardi and Masina [{\it Fract. Calc. Appl. Anal.} {\bf 21} (2018) 1156--1169]. We extend the definition of this function using the two-parameter Mittag-Leffler function. The expansions of the similarly extended sine and cosine integrals are also discussed. Numerical examples are presented to illustrate the accuracy of each type of expansion obtained.

An asymptotic expansion related to the Dickman function

We prove a conjecture of Broadurst (arXiv:1004.0519v1) on asymptotic expansions of certain polylogarithm type functions related to the Dickman function.

Uniform asymptotic expansions for the Whittaker functions $M_{κ,μ}(z)$ and $W_{κ,μ}(z)$ with $μ$ large

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…

Error bounds for the large-argument asymptotic expansions of the Hankel and Bessel functions

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise fro…

具有奇异相位的振荡积分的渐近展开

Asymptotic expansion of oscillatory integrals with singular phases

本文研究相位具有奇异性的一维振荡积分，给出其奇异性的渐近展开描述，并将类似结果推广到拉普拉斯积分。

Asymptotic Expansions of Jacobi Polynomials for Large Values of $β$ and of Their Zeros

Asymptotic approximations of Jacobi polynomials are given for large values of the $β$-parameter and of their zeros. The expansions are given in terms of Laguerre polynomials and of their zeros. The levels of accuracy of the approximations are verified by numerical examples.

$A$-Hypergeometric Distributions and Newton Polytopes

We give a bijection between a quotient space of the parameters and the space of moments for any $A$-hypergeometric distribution. An algorithmic method to compute the inverse image of the map is proposed utilizing the holonomic gradient method and an asymptotic equivalence of the map and the iterative proportional scaling. The algorithm gives a method to solve a conditional maximum likelihood estimation problem in statistics. Our interplay between the theory of hypergeometric functions and statistics gives some new formulas of $A$-hypergeometric polynomials.

关于相关正态随机变量乘积分布的渐近展开

Asymptotic expansions relating to the distribution of the product of correlated normal random variables

本文为相关正态变量乘积及其独立和的尾部分布与分位数函数导出渐近展开，并用数值实验验证其性能。

On the coefficients of the asymptotic expansion of n!

Applying a theorem of Howard for a formula recently proved by Brassesco and Méndez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion to the sequence of factorials. To our knowledge no explicit formula containing only the four basic operations was known until now.

On asymptotic expansions of oscillatory integrals with phase functions expressed by a product of positive real power function and real analytic function in one variable

In this paper, by using asymptotic expansions of oscillatory integrals with positive real power phase functions in one variable, we obtain asymptotic expansions of oscillatory integrals with phase functions expressed by a product of positive real power function and real analytic function in one variable. Moreover we show an example which we can compute all coefficients of terms in asymptotic expansions concretely.

无穷远处渐近展开的证明

Justification of an asymptotic expansion at infinity

本文考虑常微分方程组在无穷远处的渐近解族，并证明存在具有这些渐近性的精确解。