| Dewey Class |
515.5 |
| Title |
A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions ( EBook/) / by Jean-Luc Marichal, Naïm Zenaïdi. |
| Author |
Marichal, Jean-Luc |
| Added Personal Name |
Zenaïdi, Naïm |
| Other name(s) |
SpringerLink (Online service) |
| Edition statement |
1st ed. 2022. |
| Publication |
Cham : : Springer International Publishing : : Imprint: Springer, , 2022. |
| Physical Details |
XVIII, 323 p. : online resource. |
| Series |
Developments in mathematics 2197-795X ; ; 70 |
| ISBN |
9783030950880 |
| Summary Note |
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.: |
| Contents note |
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-Webster's asymptotic condition -- C. On a question raised by Webster -- D. Asymptotic behaviors and bracketing -- E. Generalized Webster's inequality -- F. On the differentiability of \sigma_g -- Bibliography -- Analogues of properties of the gamma function -- Index. |
| Mode of acces to digital resource |
Digital reproduction.- |
| Cham : |
| Springer International Publishing, |
| 2022. - |
| Mode of access: World Wide Web. System requirements: Internet Explorer 6.0 (or higher) or Firefox 2.0 (or higher). Available as searchable text in PDF format. |
| System details note |
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users). |
| Internet Site |
https://doi.org/10.1007/978-3-030-95088-0 |
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