An asymptotic expansion for a ratio of products of gamma functions
Wolfgang Buehring
推导出Γ函数乘积比的渐近展开式。它概括了一个公式，这是说丁格尔，首先证明了巴黎，最近重新考虑的奥尔弗。
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.
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Asymptotic expansions for ratios of products of gamma functions
本文导出了Γ函数乘积比的渐近展开式。
Asymptotic expansions of integral means and applications to the ratio of gamma functions
积分平均是一类重要的二元平均。本文证明了积分平均值渐近展开中系数的一种非常一般的计算方法。它基于显式求解形式为$B(A(x))=C(x)$的方程，其中$B$和$C$具有已知的渐近展开。通过计算与伽玛函数和双Γ函数有关的几个重要积分平均值，说明了所得结果。
Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function
本文利用三Γ函数$ψ'$建立了阶乘$n!$和Γ函数$Γ$的一些渐近公式和二重不等式。
Motivated by several conjectures posed in the paper " Completely monotonic degrees for a difference between the logarithmic and psi functions",we confirm in this work some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and the digamma function and we give two bounded for this degrees.
Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters
研究了当a趋于无穷大时，不完全Γ函数Gamma（-a，-z）和Gamma（-a，-z）的渐近行为.过渡区z~a需要用一致展开式来描述，此时误差函数是主要的逼近量。我们使用不完全Γ函数的整表示，并通过应用用于伽玛（a，z）和伽玛（a，z）的现有一致展开式的技术导出一致方程。并与Olver关于广义指数积分的一致展开式进行了比较.在最后一节中给出了膨胀的数值验证。
A new uniform asymptotic expansion for the incomplete gamma function $Γ(a,z)$ valid for large values of $z$ was given by the author in {\it J. Comput. Appl. Math.} {\bf 148} (2002) 323--339. This expansion contains a complementary error function of an argument measuring transition across the point $z=a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $Γ(a,a)$.
We construct asymptotic expansions for the normalised incomplete gamma function $Q(a,z)=Γ(a,z)/Γ(a)$ that are valid in the transition regions, including the case $z\approx a$, and have simple polynomial coefficients. For Bessel functions, these type of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are…
Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal
在（Boyd，Proc. R.长索克A 447（1994）609- 630），W. G. C.博伊德推导出一个复苏表示的Γ函数，利用重新制定的方法最陡下降的M。Berry和C. Howls（Berry和Howls，Proc. R. Soc. Lond. A 434（1991）657- 675）。使用这种表示法，他能够推导出Γ函数的渐近展开式的一些性质，包括明确和现实的误差界，斯托克斯不连续性的平滑过渡，以及后期系数的渐近性。本文的主要目的是修正Boyd的Rehabilitation公式，使之适用于推导Γ函数及其倒数的渐近展开的较好的误差估计。我们还证明了这些扩展的指数改进的版本完成与误差项。最后，我们提供了新的（正式的）渐近展开的系数出现在渐近级数，并比较其数值效果与早期作者的结果。
Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function
引入$r>0$级完全单调函数，证明了Barnes双Γ函数和Euler Γ函数的对数的渐近展开的展开式可产生任意正整数级的完全单调函数.
In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} Γ\left( x+1\right) =\sqrt{2πx}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left( 2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}} +R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left| R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left( 2n…
Let $K,M,N$ denote three bivariate means. In the paper, the author prove the asymptotic formulas for the gamma function have the form of% \begin{equation*} Γ\left( x+1\right) \thicksim \sqrt{2π}M\left( x+θ,x+1-θ\right) ^{K\left( x+ε,x+1-ε\right) }e^{-N\left( x+σ,x+1-σ\right) } \end{equation*}% or% \begin{equation*} Γ\left( x+1\right) \thicksim \sqrt{2π}M\left( x+θ,x+σ\right) ^{K\left( x+ε,x+1-ε\right) }e^{-M\left( x+θ,x+σ\right) } \end{equation*}% as $x\rightarrow \infty $, where $ε,θ,σ$ are fixed real numbers. This idea can be extended to the psi and polygamma functions. As examples, some new…
On Gautschi & Stirling Identities, Asymptotics and Inequalities for the Pi (or Gamma) Function
本文推导了π函数及伽马函数的Stirling型渐近公式的双侧界，并通过渐近最优的界改进了相关结果。
Resurgence in the Transition Region: The Incomplete Gamma Function
本文首次研究不完全伽马函数过渡区域渐近展开系数的 resurgence 性质，发现系数渐近行为依赖奇偶性，并利用初等工具与已知伽马函数渐近展开性质给出刻画。
On Asymptotics Of Γ_{q}(z) As q Approaching 1
本文给出了q-Γ函数在q趋近于1时的渐近主项的一个推导。该公式适用于除欧拉Γ函数极点以外的所有复平面。
In this paper we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.
On some expansions for the Euler Gamma function and the Riemann Zeta function
本文介绍了Euler Γ函数和黎曼ζ函数的几种基于下降因子的展开式。在证明中，我们使用了法迪布鲁诺公式，贝尔多项式，潜在的多项式，Mittag-Leffler多项式，导数多项式和特殊的数字（欧拉数和斯特林数的两种）。我们研究了该级数的收敛速率并给出了一些数值例子。
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。
In this paper we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function $Γ\left( { - a,λa} \right)$ with large $a$ and fixed positive $λ$, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingle's formal result regarding the exponentially improved version of the asymptot…
Asymptotic expansion of a function defined by power series
给出了由Taylor级数定义的函数的负幂级数渐近展开式存在的一个充分条件，并给出了该展开式系数的酉公式.给出了一个反正切函数的计算实例。
On Asymptotics of $q$-Gamma Functions
在本文中，我们得到了一些渐近公式的$q$ -Γ函数$Γ_{q}(z)$的$q$趋于1。
In a recent paper \cite{Temme:2021:AKH} new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a\sim b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.
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.
Parabolic Cylinder Functions: Examples of Error Bounds For Asymptotic Expansions
本文讨论了抛物柱面函数的几种渐近展开式，并给出了展开式中逼近器的误差界。特别是庞加莱{é}型展开的大值的参数$z$和大值的参数的一致的扩展被认为是。这些展开式基于Olver（1959）给出的展开式和Temme（2000）给出的这些展开式的修改。计算机代数技术用于获得表示的界限和数值计算。
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。
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.
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 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.
More accurate approximations for the Gamma function
一个由R. W.高斯分布和中心二项式系数的一些渐近展开式使我们得到了Γ函数的新的精确近似。
We consider differences between $\log Γ(x)$ and truncations of certain classical asymptotic expansions in inverse powers of $x-λ$ whose coefficients are expressed in terms of Bernoulli polynomials $B_n(λ)$, and we obtain conditions under which these differences are strictly completely monotonic. In the symmetric cases $λ=0$ and $λ=1/2$, we recover results of Sonin, Nörlund and Alzer. Also we show how to derive these asymptotic expansions using the functional equation of the logarithmic derivative of the Euler gamma function, the representation of $1/x$ as a difference $F(x+1)-F(x)$, and a back…
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.
On the expansion of the Kummer function in terms of incomplete Gamma functions
本文讨论了Kummer超几何函数在参数和变量为实值时作为一系列不完全Gamma函数的展开式。近似的Kummer函数与有限的伽玛和执行的错误进行了分析评估。它的界限推导出，无论是逐点和一致的变量，这些特点的收敛速度的系列，无论是逐点和适当的超级规范。同样的分析表明，极少数伽玛的有限和足够接近库默尔函数。结合这些结果与已知的近似方法的不完整的伽玛允许构造上，下近似的库默尔函数只使用指数，真正的权力和合理的功能。提供了说明性的例子。
An algorithm for computing the incomplete gamma function $γ^*(a,z)$ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincaré-type expansions, uniform asymptotic expansions and recurrence relations, depending on the parameter region. A relative accuracy $\sim 10^{-13}$ in the parameter region $(a,z) \in [-500,\,500] \times [-500,\,0)$ can be obtained when computing the function $γ^*(a,z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.
On some properties of special functions involving $k$-gamma and $k$-digamma functions
本研究基于$k$-伽马与$k$-双伽马函数，推导Furdui型积分的级数展开及两类特殊函数的新恒等式、展开式与不等式，并提出一个开放问题。
Jacob's ladders, Hardy-Littlewood integral (1918) and new asymptotic functional equations for Euler's Gamma function together with the tenth equivalent of the Fermat-Wiles theorem
本文基于Hardy-Littlewood积分的几乎线性增量构造了新$Γ$泛函，生成了Fermat-Wiles定理的$Γ$等价式及Euler $Γ$函数的新因式分解公式集。
An asymptotic expansion related to the Dickman function
证明了Broadurst（arXiv：1004.0519v1）关于与Dickman函数有关的某些重对数型函数的渐近展开的一个猜想.
In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen [Monotonicity properties, inequalities, and asymptotic expansions associated with the gamma function. Appl. Math. Comput. 283 (2016), 385--396.].
We present a geometric approach to the asymptotics of the Legendre polynomials $P_{k,n+1}$, based on the Szegö kernel of the Fermat quadric hypersurface, and leading to complete asymptotic expansions holding on expanding subintervals of $[-1,1]$.
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Asymptotic expansion of holonomic distributions of one complex variable
给出了单复变数完整分布芽的一般形式。
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.
A new product formula for $(z;q)_\infty$, with applications to asymptotics
本文提出一个新的$(z;q)_\infty$乘积公式，并利用该公式研究$q$趋近于1时的渐近展开问题。
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。
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