Infinite Product Representations fot Multiple Gamma Function
Michitomo Nishizawa
给出了Vignéras多重Γ函数的两种无穷乘积表示.作为这些公式的应用，导出了该函数的乘法公式。
Two kinds of infinite product representations for Vignéras multiple gamma function are presented. As an application of these formulas, a multiplication formula for the function is derived.
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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。
An Infinite Product for e^gamma via Hypergeometric Formulas for Euler's Constant, gamma
我们推导出欧拉常数γ的超几何公式。一个“副产品”的迪埃的变换是一个无穷乘积的e^伽玛涉及的二项式系数。替代的，非超几何证明使用伽马的二重积分，贝塔积分和双Γ函数的积分。（感谢P. Sebah，S。Zlobin和W. Zudilin的宝贵建议用于论文中。
We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has different scaling factor. We give necessary and sufficient conditions in terms of nonnegativity of some elementary function and more practical sufficient conditions in terms of parameters. Further, we study the representing measure in Bernstein's theorem for both equal and non-equal scaling factors. This leads to conditions on parameters under which Meijer's $G$-function or Fox's $H$-function represents an infinitely divisible probabili…
An algorithm for evaluating the Gamma function and ramifications
本文给出了一种计算任意（有理）点上的Γ函数的新算法，以及一种新的不含Euler常数和Mascheroni常数的无穷乘积表示，并由此得到了一些新的公式和不等式.
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.
Inequalities and monotonicity properties for gamma and q-gamma functions
我们证明了一些新的结果，并统一了旧的证明，涉及完全单调的表达式涉及伽玛和$q$ -Γ函数，$0 < q < 1$。这些结果中的每一个都暗示了相关概率测度的无限可分性。在少数情况下，我们能够得到简单单调性而不具有完全单调性。所有的结果导致这些功能的不平等。其中许多是由沃尔特·高茨基在1959年的一篇论文中提出的。我们表明，一些界限可以扩展到复杂的参数。
We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, associated to good rational cones. We explain how good cones are related to collections of $SL_r(\mathbb{Z})$-elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko, and the results about the usual multiple sine and elliptic gamma functions found by Narukawa.
Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{κ,μ}\left( x\right) $ function
研究 Whittaker 函数 $\mathrm{M}_{κ,μ}(x)$ 的参数一阶导数，给出其无穷级数与有限、无穷对数积分表示，并推导相关闭式约化公式。
We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, labelled by rational cones in $\mathbb{R}^r$. For $r=2,3$ we prove that the generalized multiple elliptic gamma functions enjoy a modular property determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko. The generalized multiple sine enjoy a related infinite product representation, generalizing the results of Narukawa for the ordinary multiple sine functions.
Multiple Gamma Function and Its Application to Computation of Series
多重Γ函数$Γ_n$由递归函数方程定义，作为欧拉Γ函数的推广，最初是由Kinkelin，Glaucet和Barnes在1900年左右引入的。今天，由于Conrey、Katz和Sarnak的开创性工作，人们对多重Γ函数的兴趣又恢复了。本文讨论了$Γ_n$函数的一些理论问题及其在级数求和和无穷乘积中的应用。
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。
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 …
Gautchi's ratio and the Volume of the unit ball in R^n
设Ω（n）为R^n中单位球的体积。我们将Γ函数比伽玛（x+1/2）/伽玛（x）（x>0）表示为一个无穷乘积，这使我们能够再现和/或产生各种公式和不等式，其中一些似乎是新的，涉及Omega（n-1）/Omega（n）和（Omega（n））^2/Omega（n-1）Omega（n+1）
Various considerations on hypergeometric series
Enestrom指数中的E661。这是最初出版的“Variae explanationes circa系列hypergeometricas”（1776年）。在本文中，欧拉正在寻找的渐近行为的无穷乘积是类似的Γ函数。他着眼于一些无穷乘积和积分之间的关系。他对这些无穷乘积取对数，然后用欧拉-麦克劳林求和公式展开。在第14节中，欧拉似乎重新推导了他在论文中已经证明的一些结果。然而，我看不出这些推导有什么不同。如果任何读者认为他们理解请我将不胜感激，如果你能给我发电子邮件。我目前正在研究欧拉的工作解析数论。我想了解的两个主要课题是解析函数的解析延拓和与发散级数的联系，以及Γ函数的渐近行为。
Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ ζ(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $ζ(z)$, the generalized Stieltjes constants $γ_k(q)$, the Euler constant $γ$, Euler's gamma function $Γ(q)$ and the digamma function $ψ(q)$ have many close connections on their definitions and properties. There are also many integrals, series or infinite product representations of them along the history. In this note, we try to provide a parallel story for the alternating Hurwitz zeta function (also known as the Hurwitz-type Eul…
Homogenization of nonconvex integrals with convex growth
研究了变分法中周期多重积分的均匀化问题，当被积函数在矩阵的凸集外取无穷值时，其Gamma收敛性.
New Wallis- and Catalan-Type Infinite Products for $π, e$, and $\sqrt{2+\sqrt2}$
本文将Wallis的1655型$π/2$无穷乘积推广到$(π/K)\csc(π/K)$无穷乘积，并给出了$π/4, 2, \sqrt{2+\sqrt2}, 2π/3\sqrt3,$和其它常数的Wallis型乘积.证明使用一个经典的无穷乘积公式，涉及伽玛函数。我们还扩展加泰罗尼亚的1873年的无限产品的自由基$e$加泰罗尼亚型产品$e/4,\sqrt{e}$，和$e^{3/2}/2$。这里的证明使用斯特林公式。最后，我们找到了一个类似的$e^{2/3}/\sqrt3$的Pippenger 1980年的无限产品的$e/2$，我们猜想，他们可以推广到产品的一个权力的$e^{1/K}$。
A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial
本文以导引形式系统回顾了Bohr-Mollerup定理向高阶凸函数情形的推广理论，展示其解如何继承对数伽马函数的众多经典性质，并给出一个说明性应用。
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