  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.
