A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions by

Description: A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions by Jean-Luc Marichal, Naïm Zenaïdi 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-Mollerups 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-Mollerups 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-Mollerups theorem itself, Eulers reflection formula, Gauss multiplication theorem, Stirlings 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-Mollerups theorem and to spark the interest of a large number of researchers in this beautiful theory. FORMAT Paperback LANGUAGE English CONDITION Brand New Back Cover 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-Mollerups 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-Mollerups 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-Mollerups theorem itself, Eulers reflection formula, Gauss multiplication theorem, Stirlings 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-Mollerups theorem and to spark the interest of a large number of researchers in this beautiful theory. Author Biography 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. Table of Contents Preface.- List of main symbols.- Table of contents.- Chapter 1. Introduction.- Chapter 2. Preliminaries.- Chapter 3. Uniqueness and existence results.- Chapter 4. Interpretations of the asymptotic conditions.- Chapter 5. Multiple log-gamma type functions.- Chapter 6. Asymptotic analysis.- Chapter 7. Derivatives of multiple log-gamma type functions.- Chapter 8. Further results.- Chapter 9. Summary of the main results.- Chapter 10. Applications to some standard special functions.- Chapter 11. Definining new log-gamma type functions.- Chapter 12. Further examples.- Chapter 13. Conclusion.- A. Higher order convexity properties.- B. On Krull-Websters asymptotic condition.- C. On a question raised by Webster.- D. Asymptotic behaviors and bracketing.- E. Generalized Websters inequality.- F. On the differentiability of \sigma_g.- Bibliography.- Analogues of properties of the gamma function.- Index. Feature 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 Details ISBN3030950905 Author Naïm Zenaïdi Series Developments in Mathematics Language English Year 2022 ISBN-10 3030950905 ISBN-13 9783030950903 Format Paperback Series Number 70 Publisher Springer Nature Switzerland AG Edition 1st Imprint Springer Nature Switzerland AG Place of Publication Cham Country of Publication Switzerland Pages 323 Illustrations XVIII, 323 p. Publication Date 2022-07-07 UK Release Date 2022-07-07 Edition Description 1st ed. 2022 Alternative 9783030950873 DEWEY 515.7 Audience Professional & Vocational We've got this At The Nile, if you're looking for it, we've got it. 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