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A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial
;
2024In Aequationes Mathematicae, 98 (2), p. 455–481
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Keywords :
Difference equation; higher order convexity; Bohr-Mollerup's theorem; principal indefinite sum; Gauss' limit; Euler product form; Raabe's formula; Binet's function; Stirling's formula; Gauss' multiplication formula; Euler's constant; gamma and polygamma functions
Abstract :
[en] 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.
Disciplines :
Mathematics
Author, co-author :
Marichal, Jean-Luc ;  University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Zenaïdi, Naïm;  University of Liège, Department of Mathematics, Liège, Belgium
External co-authors :
yes
Language :
English
Title :
A generalization of Bohr-Mollerup's theorem for higher order convex functions: a tutorial
Publication date :
29 March 2024
Journal title :
Aequationes Mathematicae
ISSN :
0001-9054
eISSN :
1420-8903
Publisher :
Birkhauser Verlag, Basel, Switzerland
Volume :
98
Issue :
2
Pages :
455–481
Peer reviewed :
Peer Reviewed verified by ORBi
Focus Area :
Computational Sciences
Funders :
University of Luxembourg - UL
Commentary :
This paper is a reference tutorial/summary of an open access monograph published in the Springer Developments in Mathematics. This monograph can be downloaded at the following address: https://link.springer.com/book/9783030950873
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