论 Euler Gamma 函数的定义

On the value distribution of the Riemann zeta-function and the Euler gamma-function

We prove some uniqueness results for the Riemann zeta-function and the Euler gamma-function by virtue of shared values using the value distribution theory.

On algebraic differential equations concerning the Riemann-zeta function and the Euler-gamma function

In this paper, we prove that $ζ$ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in $Γ, Γ^{(n)}, Γ^{(l)}$ over the ring of polynomials in $\mathbb{C}$, $l>n\geq 1$ are positive integers. We extended the result that $ζ$ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $Γ, Γ', Γ''$ over the field of complex numbers, which is proved by Li and Ye[7].

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…

The generalized gamma functions

In this paper, we introduce a way to generalize the Euler's gamma function as well as some related special functions. With a given polynomial in one variable $f(t)\ge 0$, we can associate a function, so-called "gamma function associated with $f$", defined by $Γ_f(s) := \int_0^\infty f^{s-1} e^{-t} dt$. This function has many features similar to the Euler's gamma function. We also present some initial results on the gamma-type functional equation for $Γ_f(s)$ in some special cases.

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.

Carlitz 模与 Euler gamma 函数的微分 Ax-Lindemann-Weierstrass 定理

The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function

本文证明了 Euler gamma 函数的 Ax-Lindemann-Weierstrass 型微分超越结果：对于模 Z 两两不同且代数于亚纯 1-周期函数域 k 的亚纯函数 ζ_i，函数 Γ(ν-ζ_i(ν)) 在 k(ν) 上微分独立。