Asymptotic expansions for ratios of products of gamma functions

Asymptotic expansions for the incomplete gamma function in the transition regions

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

In (Boyd, Proc. R. Soc. Lond. A 447 (1994) 609--630), W. G. C. Boyd derived a resurgence representation for the gamma function, exploiting the reformulation of the method of steepest descents by M. Berry and C. Howls (Berry and Howls, Proc. R. Soc. Lond. A 434 (1991) 657--675). Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities, and asymptotics for the late coefficients. The main aim of this paper is to modify the resurgence…

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.

Two asymptotic expansions for gamma function developed by Windschitl's formula

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…

Asymptotic formulas for the gamma function constructed by bivariate means

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…

关于Gautschi与Stirling恒等式，π（或伽马）函数的渐近性与不等式

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。

过渡区域中的 resurgence：不完全伽马函数

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

In this note we give a derivation of the asymptotic main term for the q-Gamma function as q approaching 1. This formula is valid on all the complex plan except at the poles of the Euler Gamma function.

The resurgence properties of the incomplete gamma function I

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

In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.

贝塞尔函数零点的一致渐近展开

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。

The resurgence properties of the Incomplete gamma function II

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

We present a sufficient condition of existence of asymptotic expansion in negative power series for a function defined by Taylor series and unitary formulas for coefficients of this expansion. An example of computing scheme for arctangent function is represented.

On Asymptotics of $q$-Gamma Functions

In this paper we derive some asymptotic formulas for the $q$-Gamma function $Γ_{q}(z)$ for $q$ tending to 1.

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters $a$, $b$ and $z$

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.

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.

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.

More accurate approximations for the Gamma function

A series transformation idea inspired by a formula of R. W. Gosper and some asymptotic expansions for the central binomial coefficients leads us to new accurate approximations for the Gamma function.

Bounds for the logarithm of the Euler gamma function and its derivatives

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…

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.

On the expansion of the Kummer function in terms of incomplete Gamma functions

The expansion of Kummer's hypergeometric function as a series of incomplete Gamma functions is discussed, for real values of the parameters and of the variable. The error performed approximating the Kummer function with a finite sum of Gammas is evaluated analytically. Bounds for it are derived, both pointwisely and uniformly in the variable; these characterize the convergence rate of the series, both pointwisely and in appropriate sup norms. The same analysis shows that finite sums of very few Gammas are sufficiently close to the Kummer function. The combination of these results with the know…

Computation of the incomplete gamma function for negative values of the argument

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.

关于含$k$-伽马函数与$k$-双伽马函数的特殊函数的若干性质

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阶梯、Hardy-Littlewood积分(1918)及Euler Gamma函数的新渐近函数方程，连同Fermat-Wiles定理的第十等价式

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

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

Monotonicity properties related to the ratio of two gamma functions

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.].

Szegö kernels and asymptotic expansions for Legendre polynomials

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 $[-1,1]$.

Asymptotic expansions of several series and their application

Asymptotic expansions of series $\sum_{k=0}^\infty ε^k(k+a)^γe^{-(k+a)^αx}$ and $\sum_{k=0}^\infty ε^k(k+a)^γ/ (x(k+a)^α+1)^μ}$ in powers of $x$ as $x\to+0$ are found, where $ε=1$ or $ε=-1$. These expansions are applied to obtain precise inequalities for Mathieu series. Keywords: Asymptotic expansion, residues, generalized Mathieu series, inequalities.

Asymptotic expansion of holonomic distributions of one complex variable

We give the general form for a germ of holonomic distribution of one complex variable.

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.

$(z;q)_\infty$的一个新乘积公式及其在渐近学中的应用

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。

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。