论文标题
Bohr-Mollerup定理的高阶凸函数的概括
A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions
论文作者
论文摘要
1922年，Harald Bohr和Johannes Mollerup使用其log-convexity属性对Euler Gamma功能建立了出色的表征。十年后，Emil Artin研究了这一结果，并将其用于使用演算的基本方法来得出伽马函数的基本特性。然后，尼古拉斯·布尔巴基（Nicolas Bourbaki）将Bohr-Mollerup的定理采用，作为他宣布伽马功能的起点。 这本开放式访问书籍沿Wolfgang Krull，Roger Webster和其他一些人发起的线条，对Bohr-Mollerup的定理进行了深远的概括，但比过去的工作要远大。特别是，这种概括表明，使用基本技术表明，非常丰富的函数频谱满足了伽马函数的几种经典特性的类似物，包括bohr-mollerup的定理本身，欧拉的反射公式，高斯的乘法定理，stirling or formula和weierstrass'an Canonals'pantrass'canonical'sullast' 通过各种示例来说明这项工作中发展的理论范围，从伽马函数本身及其变体和普遍化（q-gamma，polygamma，多个伽玛函数）到重要的特殊功能，例如Hurwitz Zeta函数以及广义STIELTJES常数。 该卷也是纪念Bohr-Mollerup定理成立100周年的机会，并激发了大量研究人员在这一美丽的理论中的兴趣。
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.
