A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions / by Jean-Luc Marichal, Naïm Zenaïdi.

Format:

Book

Author/Creator:

Marichal, Jean-Luc.

Contributor:

Zenaïdi, Naïm.

Series:

Developments in Mathematics, 2197-795X ; 70

Language:

English

Subjects (All):

Functions, Special.

Difference equations.

Functional equations.

Special Functions.

Difference and Functional Equations.

Local Subjects:

Special Functions.

Difference and Functional Equations.

Physical Description:

1 online resource (xviii, 323 pages).

Edition:

1st ed. 2022.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2022.

Language Note:

English

Summary:

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:

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.

ISBN:

3-030-95088-3

OCLC:

1335127471