A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions
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 calc...
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| Формат: | Online |
| Мова: | Англійська |
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Springer Nature
2022
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| Предмети: | |
| Онлайн доступ: | ONIX_20220713_9783030950880_14 |
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| _version_ | 1869526500264378368 |
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| author | Marichal, Jean-Luc Zenaïdi, Naïm |
| author_browse | Marichal, Jean-Luc Zenaïdi, Naïm |
| author_facet | Marichal, Jean-Luc Zenaïdi, Naïm |
| author_sort | Marichal, Jean-Luc |
| collection | Directory of Open Access Books |
| description | 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. |
| format | Online |
| id | doab-20.500.12854ir-87685 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2022 |
| publishDateRange | 2022 |
| publishDateSort | 2022 |
| publisher | Springer Nature |
| publisherStr | Springer Nature |
| record_format | ojs |
| spelling | doab-20.500.12854ir-876852025-03-15T06:06:32Z A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions Marichal, Jean-Luc Zenaïdi, Naïm 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 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. 2022-07-14T04:00:57Z 2022-07-14T04:00:57Z 2022-07-13T12:26:58Z 2022 book ONIX_20220713_9783030950880_14 OCN: 1335127471 https://library.oapen.org/handle/20.500.12657/57317 9783030950880 https://directory.doabooks.org/handle/20.500.12854/87685 eng Developments in Mathematics open access image/jpeg image/jpeg image/jpeg n/a n/a n/a https://library.oapen.org/bitstream/20.500.12657/57317/1/978-3-030-95088-0.pdf https://library.oapen.org/bitstream/20.500.12657/57317/1/978-3-030-95088-0.pdf https://library.oapen.org/bitstream/20.500.12657/57317/1/978-3-030-95088-0.pdf Springer Nature Springer International Publishing 10.1007/978-3-030-95088-0 10.1007/978-3-030-95088-0 9fa3421d-f917-4153-b9ab-fc337c396b5a Fonds National de la Recherche Luxembourg Université du Luxembourg 9486e40e-84fb-4638-808d-62ee0de510a0 629a9a64-9e76-452b-b14c-ca38638121e6 9783030950880 Springer International Publishing 323 Cham [...] [...] open access |
| spellingShingle | 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 Marichal, Jean-Luc Zenaïdi, Naïm A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions |
| title | A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions |
| title_full | A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions |
| title_fullStr | A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions |
| title_full_unstemmed | A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions |
| title_short | A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions |
| title_sort | generalization of bohr mollerup s theorem for higher order convex functions |
| topic | 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 |
| topic_facet | 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 |
| url | ONIX_20220713_9783030950880_14 |
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