About Some Theorems of Bohr-Mollerup Type

Abstract

The Gamma function of Euler %, denoted by $\Gamma$,
satisfies the functional equ\-ation $\Gamma(x+1) = x\Gamma(x)$, for all positive real $x$, is log-convex (i.e. log $\Gamma$ is convex) on $(0, \infty)$ and $\Gamma(1)=1$. A celebrated theorem of \textit{H. Bohr} and \textit{J. Mollerup} (of [3]) characterizes the Gamma function: If a real function $G$, defined on $(0,\infty)$, satisfies the functional equation $G(x+1) = xG(x)$, for all $x > 0$, is log-convex and $G(1) = 1$, then $G (x) = \Gamma(x)$, for all $x > 0$.

By the other hand, it is known that there is a function which extends to $(0,\infty)$, the sequence-function $\mbox{H}: \mathbb N^* \to \mathbb Q\cap (0,\infty)$ given by $n\mapsto \mbox{H}_n = 1 + 1/2 + 1/3 + \ldots + 1/n$, namely $H: (0, \infty) \to\mathbb R$, defined by $H(x) = \Psi(x + 1) + \gamma$, $x > 0$, where $\Psi$ is the logarithmic derivative of $\Gamma$, that is\break
$\Psi(x) = \Gamma'(x) / \Gamma(x)$ (also called the Digamma function) and
$\gamma=\lim\limits_{n\to \infty} (\mbox{H}_n -\log n)= 0.577215664\ldots$ is the constant of \textit{Euler-Mascheroni}. The statement of the basic relations $H(x) - H(x-1) = 1/x$ and $H(1) = 1$, which give us that $H(n) = 1 + 1/2 + 1/3 + \ldots + 1/n = \mbox{H}_n$, for all $n\in \mathbb N^*$ is based on the functional equation of $\Gamma$ and on the relations $\Gamma'(1) = - \gamma$, $\Gamma(1) = 1$. Moreover, $H$ results to be a concave function.

In [4], \textit{H. H. Kairies} gave a theorem which characterizes the $\Psi$ function, namely: if $g$ is defined on $(0,\infty)$, satisfies the functional equation $g(x +1) - g(x) = 1/x$, $g$ is concave and $g(1) = \gamma$, then $g(x) = \Psi(x)$, for all $x > 0$.

Exactly as in the case of the \textit{Bohr-Mollerup} theorem, this original theorem is also one that cha\-rac\-terizes a remarkable real function using a functional equation, a property of convexity (concavity) and a particular value. We call such theorems by the term of "theorems of \textit{Bohr-Mollerup} type". In this work we obtain from the theorem of \textit{Kairies} the corresponding one exactly for the function $H$, we emphasize this type of theorems, also presenting one and, in the final part, we pass to the domain of the discrete variable, showing how similar conditions can characterize some remarkable sequences (as in \cite{lupas1}-\cite{lupas5} and \cite{panaitopol}).