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 $\Delta f(x)=\ln x$ on the open half-line $(0,\infty)$ is the log-gamma function $f(x)=\ln\Gamma(x)$, where $\Delta$ denotes the classical difference operator and $\Gamma(x)$ 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 $\Delta f(x)=g(x)$, 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.

翻译：Bohr-Mollerup定理的加法版本指出，在正半轴(0,∞)上，方程Δf(x)=ln x的唯一的（至多相差一个加法常数）凸解f(x)是对数伽马函数f(x)=lnΓ(x)，其中Δ表示经典差分算子，Γ(x)表示欧拉伽马函数。在最近出版的一本开放获取著作中，作者通过考虑函数方程Δf(x)=g(x)（其中g可选自具有任意阶凸性或凹性性质的广阔而丰富的函数类）提供并阐述了对Bohr-Mollerup定理的深远推广。他们还证明了该推广产生的解f(x)满足对数伽马函数（或等价地，伽马函数）许多性质的对应版本，包括Bohr-Mollerup定理本身、Burnside公式、欧拉无穷乘积、欧拉反射公式、Gauss极限、Gauss乘法公式、Gautschi不等式、Legendre倍乘公式、Raabe公式、Stirling公式、Wallis乘积公式、Weierstrass无穷乘积以及Wendel不等式的类比形式。本文综述了这一新颖而引人入胜理论的主要结果，并提供了一个说明性应用。