The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. These results depend upon new expansions for powers of various products of classical theta functions. This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found.
The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. George Andrews says in a preface of this book, This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.'
Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.
Автор: Lisa Jacobsen Название: Analytic Theory of Continued Fractions III ISBN: 3540518304 ISBN-13(EAN): 9783540518303 Издательство: Springer Рейтинг: Цена: 23280.00 T Наличие на складе: Есть у поставщика Поставка под заказ.
Автор: Claude Brezinski Название: History of Continued Fractions and Pad? Approximants ISBN: 3642634885 ISBN-13(EAN): 9783642634888 Издательство: Springer Рейтинг: Цена: 204970.00 T Наличие на складе: Есть у поставщика Поставка под заказ. Описание: The history of continued fractions is certainly one of the longest among those of mathematical concepts, since it begins with Euclid`s algorithm for the great- est common divisor at least three centuries B.C.
Автор: Borwein Название: Neverending Fractions ISBN: 0521186498 ISBN-13(EAN): 9780521186490 Издательство: Cambridge Academ Рейтинг: Цена: 40130.00 T Наличие на складе: Есть у поставщика Поставка под заказ. Описание: This contemporary introduction to the fascinating theory of continued fractions covers a variety of applications that will interest graduates, postgraduates and researchers, as well as teachers and even amateur enthusiasts.
Автор: Karpenkov, Oleg Название: Geometry of continued fractions ISBN: 3642393675 ISBN-13(EAN): 9783642393679 Издательство: Springer Рейтинг: Цена: 69870.00 T Наличие на складе: Есть у поставщика Поставка под заказ. Описание: Geometry of Continued Fractions
Автор: Landman Bruce, Nathanson Melvyn B., Ne Etril Jaros Название: Combinatorial Number Theory: Proceedings of the "Integers Conference 2011, " Carrollton, Georgia, USA, October 26-29, 2011 ISBN: 3110280485 ISBN-13(EAN): 9783110280487 Издательство: Walter de Gruyter Рейтинг: Цена: 185890.00 T Наличие на складе: Есть у поставщика Поставка под заказ. Описание: This volume contains selected refereed papers based on lectures presented at the "Integers Conference 2011", an international conference in combinatorial number theory that was held in Carrollton, Georgia, United States in October 2011. This was the fifth Integers Conference, held bi-annually since 2003. It featured plenary lectures presented by Ken Ono, Carla Savage, Laszlo Szekely, Frank Thorne, and Julia Wolf, along with sixty other research talks. This volume consists of ten refereed articles, which are expanded and revised versions of talks presented at the conference. They represent a broad range of topics in the areas of number theory and combinatorics including multiplicative number theory, additive number theory, game theory, Ramsey theory, enumerative combinatorics, elementary number theory, the theory of partitions, and integer sequences.
Автор: Jean-Marie De Koninck, Florian Luca Название: Analytic Number Theory: Exploring the Anatomy of Integers ISBN: 0821875779 ISBN-13(EAN): 9780821875773 Издательство: Mare Nostrum (Eurospan) Рейтинг: Цена: 74850.00 T Наличие на складе: Невозможна поставка. Описание: The authors assemble a fascinating collection of topics from analytic number theory that provides an introduction to the subject with a very clear and unique focus on the anatomy of integers, that is, on the study of the multiplicative structure of the integers. Some of the most important topics presented are the global and local behavior of arithmetic functions, an extensive study of smooth numbers, the Hardy-Ramanujan and Landau theorems, characters and the Dirichlet theorem, the $abc$ conjecture along with some of its applications, and sieve methods. The book concludes with a whole chapter on the index of composition of an integer. One of this book's best features is the collection of problems at the end of each chapter that have been chosen carefully to reinforce the material. The authors include solutions to the even-numbered problems, making this volume very appropriate for readers who want to test their understanding of the theory presented in the book.
Автор: E. Grosswald Название: Representations of Integers as Sums of Squares ISBN: 1461385687 ISBN-13(EAN): 9781461385684 Издательство: Springer Рейтинг: Цена: 97820.00 T Наличие на складе: Есть у поставщика Поставка под заказ.
Автор: P. E. Jr. Conner Название: Notes on the Witt Classification of Hermitian Innerproduct Spaces over a Ring of Algebraic Integers ISBN: 0292740670 ISBN-13(EAN): 9780292740679 Издательство: Marston Book Services Рейтинг: Цена: 26390.00 T Наличие на складе: Есть у поставщика Поставка под заказ. Описание: These lectures are intended to give mathematicians at the graduate level and beyond some powerful algebraic and number theoretical tools for formulating and solving certain types of classification problems in topology.
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