On Asymptotics of $q$-Gamma Functions

Double Inequalities for Complete Monotonicity Degrees of Remainders of Asymptotic Expansions of the Gamma and Digamma Functions

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.

$(z;q)_\infty$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。

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…

A uniform asymptotic expansion for the incomplete gamma functions revisited

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 $Γ(a,a)$.

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。

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…

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.

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…

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.

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…

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。

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。

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.].

Contributions to the Theory of the Barnes Function

This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma function and Glaisher's constant are also discussed.

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…

Gamma conjecture via mirror symmetry

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's $Γ$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argumen…

Interpolating the Derivatives of the Gamma Function

We consider a function $G(λ, z)$, entire in $λ$, which interpolates the derivatives of the Gamma function in the sense that $G(m, z) = Γ^{(m)}(z)$ for any integer $m \geq 0$ and we calculate the asymptotics of $G(λ, z)$ as $λ\to +\infty$.

Exactification of Stirling's Approximation for the Logarithm of the Gamma Function

Exactification is the process of obtaining exact values of a function from its complete asymptotic expansion. Here Stirling's approximation for the logarithm of the gamma function or $\ln Γ(z)$ is derived completely whereby it is composed of the standard leading terms and an asymptotic series that is generally truncated. Nevertheless, to obtain values of $\ln Γ(z)$, the remainder must undergo regularization. Two regularization techniques are then applied: Borel summation and Mellin-Barnes (MB) regularization. The Borel-summed remainder possesses an infinite convergent sum of exponential integr…

An asymptotic approximation for the Riemann zeta function revisited

We revisit a representation for the Riemann zeta function $ζ(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics of the incomplete gamma function produces an asymptotic-like expansion for $ζ(s)$ on the critical line $s=1/2+it$ as $t\to+\infty$. The main term involves the original Dirichlet series smoothed by a complementary error function of appropriate argument together with a series of correction terms. It is the aim here to present these correction terms in a more user-…

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class G_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by G_F Ch(E_i) for an exceptional collection {E_i} in the derived category of …

关于对数伽马函数和黎曼-西格尔θ函数渐近近似的渐近逼近

On asymptotic approximations to the log-Gamma and Riemann-Siegel theta functions

本文研究了复数z在右半平面中对数伽马函数lnΓ(z)渐近逼近的误差界，改进了早期的界，同时讨论了对黎曼-西格尔θ函数渐近逼近的误差界，并通过引入指数小项提高了对θ(t)的逼近精度。

On the asymptotic expansion of $Γ(x)$, Lagrange's inversion theorem and the Stirling coefficients

We show how the asymptotic expansion for the gamma function $Γ(x)$, similar to that obtained by Boyd [Proc. Roy. Soc. London A447 (1994) 609--630], can be obtained by using a form of Lagrange's inversion theorem with a remainder. A (possibly) new closed-form representation for the Stirling coefficients is given.

Scaled Asymptotics For Some $q$-Series

In this work we investigate the asymptotics for Euler's $q$-Exponential $E_{q}(z)$, $q$-Gamma function $Γ_{q}(z)$, Ramanujan's function $A_{q}(z)$, Jackson's $q$-Bessel function $J_ν^{(2)}$(z;q) of second kind, Stieltjes-Wigert orthogonal polynomials $S_{n}(x;q)$ and $q$-Laguerre polynomials $L_{n}^{(α)}(x;q)$ as $q$ approaching 1.

Some Monotonicity Properties of Gamma and $q$-gamma Functions

We prove some properties of completely monotonic functions and apply them to obtain results on gamma and $q$-gamma functions.

Generalizations of Some Inequalities for the $p$-Gamma, $q$-Gamma and $k$-Gamma Functions

In this paper, we present and prove some generalizations of some inequalities for the $p$-Gamma, $q$-Gamma and $k$-Gamma functions. Our approach makes use of the series representations of the psi, $p$-psi, $q$-psi and $k$-psi functions.

Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions

Kummer's Fourier series for the log gamma function is well known, having been discovered in 1847. In this paper we develop a corresponding Fourier series for the logarithm of the Barnes double gamma function (and the method may be easily extended to higher order multiple gamma functions). Some applications of these Fourier series are explored.

New proofs of the duplication and multiplication formulae for the gamma and the Barnes double gamma functions

New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for the double gamma function are also reported. This paper also refers to some connections with the Stieltjes constants.

关于函数方程 $f(x+1)=g(x)f(x)$ 的评注及 gamma 函数的一个唯一性定理

Remarks on the functional equation $f(x+1)=g(x)f(x)$ and a uniqueness theorem for the gamma function

本文研究函数方程 $f(x+1)=g(x)f(x)$ 的唯一性条件，减弱并推广 Webster 的渐近假设，证明二阶对数凸解的唯一性，并据此给出 gamma 函数的新刻画。

Some completely monotonic functions involving the $q$-gamma function

We present some completely monotonic functions involving the $q$-gamma function that are inspired by their analogues involving the gamma function.

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。

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.

On the definition of higher Gamma functions

We generalize our previous new definition of Euler Gamma function to higher Gamma functions. With this unified approach, we characterize Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson $q$-Gamma function, and Nishizawa higher $q$-Gamma functions. This approach extends to more general functional equations. This generalization reveals the multiplicative group structure of solutions of the functional equation that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. In this new app…

A completely monotonic function involving the gamma and tri-gamma functions

In the paper the author provides necessary and sufficient conditions on $a$ for the function $\frac{1}{2}\ln(2π)-x+\bigl(x-\frac{1}{2}\bigr)\ln x-\lnΓ(x)+\frac1{12}{ψ'(x+a)}$ and its negative to be completely monotonic on $(0,\infty)$, where $a\ge0$ is a real number, $Γ(x)$ is the classical gamma function, and $ψ(x)=\frac{Γ'(x)}{Γ(x)}$ is the di-gamma function. As applications, some known results and new inequalities are derived.

Properties of Ultra Gamma Function

In this paper we study the integral of type \[_{δ,a}Γ_{ρ,b}(x) =Γ(δ,a;ρ,b)(x)=\int_{0}^{\infty}t^{x-1}e^{-\frac{t^δ}{a}-\frac{t^{-ρ}}{b}}dt.\] Different authors called this integral by different names like ultra gamma function, generalized gamma function, Kratzel integral, inverse Gaussian integral, reaction-rate probability integral, Bessel integral etc. We prove several identities and recurrence relation of above said integral, we called this integral as Four Parameter Gamma Function. Also we evaluate relation between Four Parameter Gamma Function, p-k Gamma Function and Classical Gamma Func…

Inequalities for the Psi and $k$-Gamma Functions

In this paper, the authors establish some inequalities involving the Psi and $k$-Gamma functions. The procedure utilizes some monotonicity properties of some functions associated with the Psi and $k$-Gamma functions.

Bounds for the ratio of two gamma functions--From Gautschi's and Kershaw's inequalities to completely monotonic functions

In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma functions, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions, some new bounds for the ratio of two gamma functions and divided differences of polygamma functions, and related monotonicity results.

Bounds for the ratio of two gamma functions--From Wendel's and related inequalities to logarithmically completely monotonic functions

In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, several complete monotonicity of functions involving ratios of two gamma or $q$-gamma functions, and necessary and sufficient conditions for functions involving ratios of two gamma or $q$-gamma functions to be logarithmically completely monotonic.

Some inequalities for the $q$-Extension of the Gamma Function

In this paper, the authors establish some inequalities involving the $q$-extension of the classical Gamma function. These inequalities provide bounds for certain ratios of the $q$-extended Gamma function. The procedure makes use of geometric convexity and monotonicy properties of certain functions associated with the $q$-extended Gamma function.

Some exponential and trigonometric integrals involving the log gamma function

This paper considers some integrals where the integrand comprises the log gamma function or the digamma function multiplied by exponential or trigonometric functions.

GammaCHI：用于伽马和卡方累积分布函数（中心和非中心）倒置与计算的软件包

GammaCHI: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

GammaCHI软件包提供可靠且精确的非中心累积分布函数倒置算法，支持伽马、误差函数及相关函数的计算与倒置。

Integral representation of some functions related to the Gamma function

We prove that the functions Phi(x)=[Gamma(x+1)]^{1/x}(1+1/x)^x/x and log Phi(x) are Stieltjes transforms.

受 Ramanujan 启发的渐近展开

An asymptotic expansion inspired by Ramanujan

本文验证了 Ramanujan 笔记本中关于一个和式渐近于 ln(x)+γ 的论断在 n=2 时正确，并讨论了误差项阶数及 n=1 情况的不同正确推广。