Topics And Methods In Q Series
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Topics And Methods In Q Series
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Author : James Mc Laughlin
language : en
Publisher: World Scientific
Release Date : 2017-09-22
Topics And Methods In Q Series written by James Mc Laughlin and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-22 with Mathematics categories.
The book provides a comprehensive introduction to the many aspects of the subject of basic hypergeometric series. The book essentially assumes no prior knowledge but eventually provides a comprehensive introduction to many important topics. After developing a treatment of historically important topics such as the q-binomial theorem, Heine's transformation, the Jacobi triple product identity, Ramanujan's 1-psi-1 summation formula, Bailey's 6-psi-6 summation formula and the Rogers-Fine identity, the book goes on to delve more deeply into important topics such as Bailey- and WP-Bailey pairs and chains, q-continued fractions, and mock theta functions. There are also chapters on other topics such as Lambert series and combinatorial proofs of basic hypergeometric identities.The book could serve as a textbook for the subject at the graduate level and as a textbook for a topic course at the undergraduate level (earlier chapters). It could also serve as a reference work for researchers in the area.
Topics And Methods In Q Series
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Author : James Mc Laughlin
language : en
Publisher: World Scientific Publishing Company
Release Date : 2017-01-11
Topics And Methods In Q Series written by James Mc Laughlin and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-11 with Hypergeometric series categories.
The book provides a comprehensive introduction to the many aspects of the subject of basic hypergeometric series. The book essentially assumes no prior knowledge but eventually provides a comprehensive introduction to many important topics. After developing a treatment of historically important topics such as the q-binomial theorem, Heine's transformation, the Jacobi triple product identity, Ramanujan's 1-psi-1 summation formula, Bailey's 6-psi-6 summation formula and the Rogers-Fine identity, the book goes on to delve more deeply into important topics such as Bailey- and WP-Bailey pairs and chains, q-continued fractions, and mock theta functions. There are also chapters on other topics such as Lambert series and combinatorial proofs of basic hypergeometric identities. The book could serve as a textbook for the subject at the graduate level and as a textbook for a topic course at the undergraduate level (earlier chapters). It could also serve as a reference work for researchers in the area.
An Invitation To Q Series
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Author : Hei-Chi Chan
language : en
Publisher: World Scientific
Release Date : 2011
An Invitation To Q Series written by Hei-Chi Chan and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
The aim of this lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers–Ramanujan continued fraction — a result that convinced G H Hardy that Ramanujan was a “mathematician of the highest class”, and (2) what G H Hardy called Ramanujan's “Most Beautiful Identity”. This book covers a range of related results, such as several proofs of the famous Rogers–Ramanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series.
Jacobsthal Sums
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Author : Bogdan Nica
language : en
Publisher: World Scientific
Release Date : 2025-05-27
Jacobsthal Sums written by Bogdan Nica and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-05-27 with Mathematics categories.
The focus of this monograph is on the Jacobsthal sums of the title. These are quadratic character sums with polynomial arguments of a certain simple form. In addition to studying Jacobsthal sums on their own, the monograph explores their role in several topics of number-theoretical interest. A prominent theme is their use in counting solutions to equations over prime fields. Another aim is to construct representations of primes as sums of squares using Jacobsthal sums. Finally, Jacobsthal sums are applied to evaluate other quadratic character sums with polynomial arguments.This text is self-contained, with minimal technical prerequisites. We have strived for an engaging exposition, incorporating numerous examples and applications, carefully selected exercises, and historical notes and perspectives. Complete solutions to all exercises are provided.This monograph should be of interest to researchers studying character sums or counting solutions to equations over finite fields, as well as to graduate students and advanced undergraduates interested in these topics.
Frontiers In Orthogonal Polynomials And Q Series
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Author : M Zuhair Nashed
language : en
Publisher: World Scientific
Release Date : 2018-01-12
Frontiers In Orthogonal Polynomials And Q Series written by M Zuhair Nashed and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-12 with Mathematics categories.
This volume aims to highlight trends and important directions of research in orthogonal polynomials, q-series, and related topics in number theory, combinatorics, approximation theory, mathematical physics, and computational and applied harmonic analysis. This collection is based on the invited lectures by well-known contributors from the International Conference on Orthogonal Polynomials and q-Series, that was held at the University of Central Florida in Orlando, on May 10-12, 2015. The conference was dedicated to Professor Mourad Ismail on his 70th birthday.The editors strived for a volume that would inspire young researchers and provide a wealth of information in an engaging format. Theoretical, combinatorial and computational/algorithmic aspects are considered, and each chapter contains many references on its topic, when appropriate.
Analytic And Combinatorial Number Theory The Legacy Of Ramanujan Contributions In Honor Of Bruce C Berndt
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Author : George E Andrews
language : en
Publisher: World Scientific
Release Date : 2024-08-19
Analytic And Combinatorial Number Theory The Legacy Of Ramanujan Contributions In Honor Of Bruce C Berndt written by George E Andrews and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-08-19 with Mathematics categories.
This volume reflects the contributions stemming from the conference Analytic and Combinatorial Number Theory: The Legacy of Ramanujan which took place at the University of Illinois at Urbana-Champaign on June 6-9, 2019. The conference included 26 plenary talks, 71 contributed talks, and 170 participants. As was the case for the conference, this book is in honor of Bruce C Berndt and in celebration of his mathematics and his 80th birthday.Along with a number of papers previously appearing in Special Issues of the International Journal of Number Theory, the book collects together a few more papers, a biography of Bruce by Atul Dixit and Ae Ja Yee, a preface by George Andrews, a gallery of photos from the conference, a number of speeches from the conference banquet, the conference poster, a list of Bruce's publications at the time this volume was created, and a list of the talks from the conference.
The Power Of Q
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Author : Michael D. Hirschhorn
language : en
Publisher: Springer
Release Date : 2017-08-08
The Power Of Q written by Michael D. Hirschhorn and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-08-08 with Mathematics categories.
This unique book explores the world of q, known technically as basic hypergeometric series, and represents the author’s personal and life-long study—inspired by Ramanujan—of aspects of this broad topic. While the level of mathematical sophistication is graduated, the book is designed to appeal to advanced undergraduates as well as researchers in the field. The principal aims are to demonstrate the power of the methods and the beauty of the results. The book contains novel proofs of many results in the theory of partitions and the theory of representations, as well as associated identities. Though not specifically designed as a textbook, parts of it may be presented in course work; it has many suitable exercises. After an introductory chapter, the power of q-series is demonstrated with proofs of Lagrange’s four-squares theorem and Gauss’s two-squares theorem. Attention then turns to partitions and Ramanujan’s partition congruences. Several proofs of these are given throughout the book. Many chapters are devoted to related and other associated topics. One highlight is a simple proof of an identity of Jacobi with application to string theory. On the way, we come across the Rogers–Ramanujan identities and the Rogers–Ramanujan continued fraction, the famous “forty identities” of Ramanujan, and the representation results of Jacobi, Dirichlet and Lorenz, not to mention many other interesting and beautiful results. We also meet a challenge of D.H. Lehmer to give a formula for the number of partitions of a number into four squares, prove a “mysterious” partition theorem of H. Farkas and prove a conjecture of R.Wm. Gosper “which even Erdős couldn’t do.” The book concludes with a look at Ramanujan’s remarkable tau function.
College Of Engineering
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Author : University of Michigan. College of Engineering
language : en
Publisher: UM Libraries
Release Date : 1992
College Of Engineering written by University of Michigan. College of Engineering and has been published by UM Libraries this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Engineering schools categories.
Analytic Number Theory Modular Forms And Q Hypergeometric Series
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Author : George E. Andrews
language : en
Publisher: Springer
Release Date : 2018-02-01
Analytic Number Theory Modular Forms And Q Hypergeometric Series written by George E. Andrews and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-02-01 with Mathematics categories.
Gathered from the 2016 Gainesville Number Theory Conference honoring Krishna Alladi on his 60th birthday, these proceedings present recent research in number theory. Extensive and detailed, this volume features 40 articles by leading researchers on topics in analytic number theory, probabilistic number theory, irrationality and transcendence, Diophantine analysis, partitions, basic hypergeometric series, and modular forms. Readers will also find detailed discussions of several aspects of the path-breaking work of Srinivasa Ramanujan and its influence on current research. Many of the papers were motivated by Alladi's own research on partitions and q-series as well as his earlier work in number theory. Alladi is well known for his contributions in number theory and mathematics. His research interests include combinatorics, discrete mathematics, sieve methods, probabilistic and analytic number theory, Diophantine approximations, partitions and q-series identities. Graduate students and researchers will find this volume a valuable resource on new developments in various aspects of number theory.
An Introduction To Basic Fourier Series
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Author : Sergei Suslov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
An Introduction To Basic Fourier Series written by Sergei Suslov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
It was with the publication of Norbert Wiener's book ''The Fourier In tegral and Certain of Its Applications" [165] in 1933 by Cambridge Univer sity Press that the mathematical community came to realize that there is an alternative approach to the study of c1assical Fourier Analysis, namely, through the theory of c1assical orthogonal polynomials. Little would he know at that time that this little idea of his would help usher in a new and exiting branch of c1assical analysis called q-Fourier Analysis. Attempts at finding q-analogs of Fourier and other related transforms were made by other authors, but it took the mathematical insight and instincts of none other then Richard Askey, the grand master of Special Functions and Orthogonal Polynomials, to see the natural connection between orthogonal polynomials and a systematic theory of q-Fourier Analysis. The paper that he wrote in 1993 with N. M. Atakishiyev and S. K Suslov, entitled "An Analog of the Fourier Transform for a q-Harmonic Oscillator" [13], was probably the first significant publication in this area. The Poisson k~rnel for the contin uous q-Hermite polynomials plays a role of the q-exponential function for the analog of the Fourier integral under considerationj see also [14] for an extension of the q-Fourier transform to the general case of Askey-Wilson polynomials. (Another important ingredient of the q-Fourier Analysis, that deserves thorough investigation, is the theory of q-Fourier series.