Overview
- This book is open access, which means that you have free and unlimited access
- Gives a far-reaching generalization of the famous Bohr-Mollerup theorem from 1922
- Provides a unified setting for the investigation of special functions
- Shows that many properties of the gamma function have counterparts for a wide variety of functions
Part of the book series: Developments in Mathematics (DEVM, volume 70)
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About this book
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.
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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Keywords
- Difference Equation
- Higher Order Convexity
- Bohr-Mollerup's Theorem
- Principal Indefinite Sums
- Gauss' Limit
- Euler Product Form
- Raabe's Formula
- Binet's Function
- Stirling's Formula
- Euler's Infinite Product
- Euler's Reflection Formula
- Weierstrass' Infinite Product
- Gauss Multiplication Formula
- Euler's Constant
- Gamma Function
- Polygamma Functions
- Hurwitz Zeta Function
- Generalized Stieltjes Constants
Table of contents (13 chapters)
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Front Matter
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Back Matter
Authors and Affiliations
About the authors
Jean-Luc Marichal is an Associate Professor of Mathematics at the University of Luxembourg. He completed his PhD in Mathematics in 1998 at the University of Liège (Belgium) and has published about 100 journal papers on aggregation function theory, functional equations, non-additive measures and integrals, conjoint measurement theory, cooperative game theory, and system reliability theory.
Naïm Zenaïdi is a Senior Teaching and Outreach Assistant in the Department of Mathematics at the University of Liège (Belgium). He completed his PhD in Mathematics in 2013 at the University of Brussels (ULB, Belgium) in the field of differential geometry.
Naïm Zenaïdi is a Senior Teaching and Outreach Assistant in the Department of Mathematics at the University of Liège (Belgium). He completed his PhD in Mathematics in 2013 at the University of Brussels (ULB, Belgium) in the field of differential geometry.
Bibliographic Information
Book Title: A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions
Authors: Jean-Luc Marichal, Naïm Zenaïdi
Series Title: Developments in Mathematics
DOI: https://doi.org/10.1007/978-3-030-95088-0
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s) 2022
Hardcover ISBN: 978-3-030-95087-3Published: 07 July 2022
Softcover ISBN: 978-3-030-95090-3Published: 07 July 2022
eBook ISBN: 978-3-030-95088-0Published: 06 July 2022
Series ISSN: 1389-2177
Series E-ISSN: 2197-795X
Edition Number: 1
Number of Pages: XVIII, 323
Topics: Special Functions, Difference and Functional Equations