论文标题
Bohr-Mollerup的高阶凸函数的概括：教程
A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial
论文作者
论文摘要
Bohr-Mollerup在其添加版本中的出色定理指出，在公开的半行$（0，\ infty）上，唯一（达到加性常数）凸面$ f（x）$ for equienta $γ（x）$表示欧拉伽马功能。在最近出版的一本开放式访问书中，作者通过考虑功能方程$ΔF（x）= g（x）$提供并说明了Bohr-Mollerup定理的深远概括，其中可以从具有任何顺序或任何顺序的属性的广泛和丰富的功能中选择$ g $。他们还表明，由于这种概括而产生的解决方案$ f（x）$满足了log-gamma功能的许多属性（或等效地，伽马功能），包括bohr-mollerup定理的类似物，伯恩赛德（Burnside）的burnside公式，欧拉（Euler Legendre的重复配方，Raabe的配方，Stirling的配方，Wallis的产品配方，Weiersstrass的无限产品以及Wendel对伽马功能的不平等。在本文中，我们回顾了这一新的有趣理论的主要结果，并提供了说明性的应用。
In its additive version, Bohr-Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution $f(x)$ to the equation $Δf(x)=\ln x$ on the open half-line $(0,\infty)$ is the log-gamma function $f(x)=\lnΓ(x)$, where $Δ$ denotes the classical difference operator and $Γ(x)$ denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr-Mollerup's theorem by considering the functional equation $Δf(x)=g(x)$, where $g$ can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions $f(x)$ arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr-Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application.
