高阶凸函数的Bohr-Mollerup定理的推广
A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions
摘要：
1922年，Harald Bohr和Johannes Mollerup利用对数凸性建立了欧拉Γ函数的一个显著特征。十年后，埃米尔·阿廷研究了这一结果，并利用微积分的初等方法导出了Γ函数的基本性质。玻尔-穆勒鲁普定理随后被尼古拉·布尔巴基采用，作为他阐述Γ函数的出发点。这本开放获取的书发展了一个深远的推广玻尔Mollerup定理高阶凸函数，沿线发起的沃尔夫冈Krull，罗杰韦伯斯特，和其他一些人，但要远远超过过去的工作。特别地，这个推广使用初等技巧表明，非常丰富的函数谱满足Γ函数的几个经典性质的类似物，包括玻尔-穆勒鲁普定理本身，欧拉反射公式，高斯乘法定理，斯特林公式和维尔斯特拉斯标准因子分解。在这项工作中开发的理论的范围说明通过各种例子，从Γ函数本身及其变体和概括（q-伽马，polygamma，多Γ函数）的重要特殊函数，如赫维茨zeta函数和广义Stieltjes常数。这卷也是一个机会，以纪念100周年的玻尔-Mollerup定理，并引发了大量的研究人员在这个美丽的理论的兴趣。
英文摘要：
In 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
