Reference : A generalization of Bohr-Mollerup's theorem for higher order convex functions
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A generalization of Bohr-Mollerup's theorem for higher order convex functions
Marichal, Jean-Luc mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)]
Zenaïdi, Naïm mailto [University of Liège, Department of Mathematics, Liège, Belgium]
Developments in Mathematics, Vol. 70
XVIII, 323
[en] Difference equation ; Higher order convexity ; Bohr-Mollerup-Artin's theorem ; Krull-Webster's theory ; Generalized Stirling's formula ; Generalized Stirling's constant ; Generalized Euler's constant ; Euler's reflection formula ; Euler's infinite product ; Weierstrass' infinite product ; Gauss' multiplication theorem ; Gauss' digamma theorem ; Raabe's formula ; Wallis's product formula ; Fontana-Mascheroni's series ; Barnes G-function ; Hurwitz zeta function ; Gamma-related function ; Multiple gamma-type function ; Generalized Stieltjes constant
[en] 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.
University of Luxembourg - UL ; Fonds National de la Recherche - FnR
FNR RESCOM: SCIENTIFIC MONOGRAPHS > Project Nr 16552440 > Jean-Luc Marichal > 2022
Researchers ; Professionals ; Students
The original publication is available at
License: CC BY
Series ISSN: 1389-2177
Series E-ISSN: 2197-795X

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