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.
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