Integral representation of some functions related to the Gamma function

An introduction to the Barnes double gamma function with an application to an integral involving the cotangent function

We show a direct approach to Kinkelin's integral involving the cotangent function.

Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function

In the paper, the authors establish integral representations of some functions related to the remainder of Burnside's formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions.

Inverses of gamma functions

Euler's Gamma function $Γ$ either increases or decreases on intervals between two consequtive critical points. The inverse of $Γ$ on intervals of increase is shown to have an extension to a Pick-function and similar results are given on the intervals of decrease, thereby answering a question by Uchiyama. The corresponding integral representations are described. Similar results are obtained for a class of entire functions of genus 2, and in particular integral representations for the double gamma function and the $G$-function of Barnes are found.

An Infinite Product for e^gamma via Hypergeometric Formulas for Euler's Constant, gamma

We derive hypergeometric formulas for Euler's constant, gamma. A "by-product" of Thomae's transformation is an infinite product for e^gamma involving the binomial coefficients. Alternate, non-hypergeometric proofs use a double integral for gamma, the beta integral, and an integral for the digamma function. (I thank P. Sebah, S. Zlobin, and W. Zudilin for valuable suggestions used in the paper.)

The modular properties and the integral representations of the multiple elliptic gamma functions

We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's transformation, and that of the elliptic gamma function was provided by Felder and Varchenko. In this paper we deal with the multiple sine functions, since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite product. We also derive integral representations of the …

Exponential Polynomials, Stirling Numbers,and Evaluation of Some Gamma Integrals

This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.

利用Beta积分加速超几何函数以推导π和Gamma函数值的新无穷级数

Accelerating the Hypergeometric Function with the Beta Integral to Derive New Infinite Series for $π$ and Values of the Gamma Function

本文通过beta积分加速超几何函数$2 F 1$，推导出$π$和Gamma函数值的新无穷级数，并基于初等反三角函数得出每项增加2.5位数字的紧凑类Ramanujan公式。

一系列涉及上不完全Gamma函数的定积分

A series of definite integrals involving upper incomplete Gamma functions

本文利用概率论推导出涉及上不完全Gamma函数的一系列定积分之和的表达式，其中正态方差混合分布起关键作用，并给出了第一类Bessel函数与Struve函数导数的新求和公式。

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。

Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters

We consider the asymptotic behavior of the incomplete gamma functions gamma(-a,-z) and Gamma(-a,-z) as a goes to infinity. Uniform expansions are needed to describe the transition area z~a in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform equation by applying techniques used for the existing uniform expansions for gamma(a,z) and Gamma(a,z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final s…

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.

The Fourier series of the log-Barnes function

In this paper we determine the Fourier series expansion of the log-Barnes function. This is the analogue of the classical result of Kummer and Malmsten. Applying this expansion we get some integrals similar to the Espinosa-Moll log-Gamma integrals with respect to $\log G$. During the course of the paper some interesting log-Gamma integrals and a log-harmonic sum are evaluated.

Gamma函数的双参数变形、相关函数及若干不等式

A two-parameter deformation of the gamma function, associated functions, and some related inequalities

本文提出一种新的双参数Gamma函数变形，推广了现有Gamma型函数，并研究了其参数相关性质及相应的beta和polygamma函数的不等式。

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.].

Logarithmically completely monotontic functions related the $q-$gamma and the $q-$digamma functions with applications

In this paper we present several new classes of logarithmically completely monotonic functions. Our functions have in common that they are defined in terms of the $q-$gamma and $q-$digamma functions. As an applications of this results, some inequalities for the $q-$gamma and the $q-$digamma functions are established. Some of the given results generalized theorems due to Alzer and Berg and C.-P. Chen and F. Qi.

Gamma函数的超椭圆值

Hyperelliptic values of the Gamma function

本文提出一种不使用椭圆K函数，而利用超椭圆积分定义的常数或周期来计算Gamma函数在特定有理点处值的方法。

抛物线梅林变换：伽马和泽塔积分表示

The Parabolic Mellin Transform: Gamma and Zeta Integral Representations

本文介绍了抛物线梅林变换（PMT），并通过其对高斯核的处理，得到经典汉克尔表示的抛物线轮廓形式，并利用该变换推导了Hurwitz泽塔函数、黎曼泽塔函数和狄利克雷eta函数的积分表示，同时提供了Gamma型和泽塔型特殊函数的统一变换字典，并将黎曼假设和林德洛夫假设重新表述为抛物线轮廓积分零点和增长的等价形式。

Pick Functions Related to the Multiple Gamma Functions of order $n$

Let $G_n$ be the Barnes multiple Gamma function of order $n$ and the function $f_n(z)$ be defined as \begin{align*} f_n(z)=\dfrac{\log G_n(z+1)}{z^n\Log z},\quad z\in \mathbb{C}\setminus (-\infty,0]. \end{align*} In this work, a conjecture to find the Stieltjes representation is proposed such that $f_n(z)$ is a Pick function. The conjecture is established for the particular case $n=3$ by examining the properties of $f_3(z)$.

The integrals in Gradshteyn and Ryzhik. Part 10: the digamma function

Many integrals in the classical table by Gradshteyn and Ryzhik can be evaluated in terms of the digamma function (= the logarithmic derivative of the gamma function). Some of them are presented here.

Bounds for singular fractional integrals and related Fourier integral operators

We prove sharp L^p-L^q endpoint bounds for singular fractional integral operators and related Fourier integral operators, under the nonvanishing rotational curvature assumption.

Homogenization of nonconvex integrals with convex growth

We study homogenization by Gamma-convergence of periodic multiple integrals of the calculus of variations when the integrand can take infinite values outside of a convex set of matrices.

On some Definite Integrals involving the Hurwitz Zeta function

We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, log Gamma(q) and log sin(q).