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

完成Meijer--Barnes $K$-函数：排除情形 $N<0$

Completing the Meijer--Barnes $K$-function: the excluded case $N<0$

本文补全了Meijer--Barnes $K$-函数在 $N<0$ 情形下的定义，证明了积分收敛性，刻画了四类容许围道及其关系，推导了留数展开和变换律，并提供了数值验证。

Some applications of the Stieltjes constants

In this paper we present some applications of the Stieltjes constants including, for example, new derivations of Binet's formulae for the log gamma function and the evaluation of some integrals related to the Barnes multiple gamma functions.

Representations for the derivative at zero and finite parts of the Barnes zeta function

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.

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 $f_3(z)$.

基于Mellin-Barnes和Borel-Laplace理论的指示性阴影微积分复兴分析框架

A Resurgent Analytic Framework for Indicial Umbral Calculus via Mellin-Barnes and Borel-Laplace Theories

通过Mellin-Barnes积分、Borel-Laplace求和与复兴分析，为指示性阴影微积分建立严格分析基础，将阴影算子提升为连续线性泛函，揭示形式代数恒等式为全局解析对应的局部展开。

Bilateral zeta functions and their applications

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversio…

Generalizations of Lerch's formula by Barnes' multiple zeta functions

The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2π}}{Γ(x)},\quad \textrm{Re}(x)>0, $$ where $Γ(x)$ is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product $$ \prod_{n=1}^\infty n=\sqrt{2π} $$ (in the sense of zeta regularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\fracπ{2}$$ (Wallis' formula in 1656), respectively.

Nielsen's beta function and some infinitely divisible distributions

We show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form $xf(x)$, where $f$ is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.

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.

Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function $Γ_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the $Γ_n$ function and their applications to summation of series and infinite products.

Ratios of Entire functions and generalized Stieltjes functions

Monotonicity properties of the ratio $$ \log \frac{f(x+a_1)\cdots f(x+a_n)}{f(x+b_1)\cdots f(x+b_n)}, $$ where $f$ is an entire function are investigated. Earlier results for Euler's gamma function and other entire functions of genus 1 are generalised to entire functions of genus $p$ with negative zeros. Derivatives of order comparable to $p$ of the expression above are related to generalised Stieltjes functions of order $p+1$. Our results are applied to the Barnes multiple gamma functions. We also show how recent results on the behaviour of Euler's gamma function on vertical lines can be shar…

On the h-function

The paper is devoted to study the $H$-function defined by the Mellin-Barnes integral $$H^{m,n}_{\thinspace p,q}(z)={\frac1{2πi}}\int_{\Lss} \HHs^{m,n}_{\thinspace p,q}(s)z^{-s}ds,$$ where the function $\HH^{m,n}_{\thinspace p,q}(s)$ is a certain ratio of products of Gamma functions with the argument $s$ and the contour $\LL$ is specially chosen. The conditions for the existence of $H^{m,n}_{\thinspace p,q}(z)$ are discussed and explicit power and power-logarithmic series expansions of $H^{m,n}_{p,q}(z)$ near zero and infinity are given. The obtained results define more precisely the known resu…

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…

Meijer $G$-函数的推广

Extending the Meijer $G$-function

本研究通过用Barnes双伽马函数替换Meijer $G$-函数定义中的欧拉伽马函数，提出一类更具一般性的新特殊函数$K$-函数，研究其解析与变换性质，建立其与随机过程、分数阶拉普拉斯算子相关函数的关联，并证明推广的Kilbas-Saigo函数是其特例。

从双曲到复欧拉积分

From hyperbolic to complex Euler integrals

研究双曲超几何积分退化为复超几何函数的过程，通过积分核一致有界性证明单变量双曲beta积分和锥函数退化为复平面上的二维积分。

Mizuno 乘积公式的交替推广与修正 gamma 函数

An alternating Mizuno formula, modified gamma functions, and double sine special values

本文建立了 Mizuno 乘积公式的完整交替推广，引入修正 gamma 函数的交替版本，并推导出 Wallis 乘积、Kurokawa--Wakayama 型公式及多重交替 gamma 函数的性质，最后给出 Shintani 双正弦函数交替模拟的算术二分法。