An Invitation to the Rogers-Ramanujan Identities

An Invitation to the Rogers-Ramanujan Identities
Author: Andrew V. Sills
Publisher: CRC Press
Total Pages: 263
Release: 2017-10-16
Genre: Mathematics
ISBN: 1351647962

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The Rogers--Ramanujan identities are a pair of infinite series—infinite product identities that were first discovered in 1894. Over the past several decades these identities, and identities of similar type, have found applications in number theory, combinatorics, Lie algebra and vertex operator algebra theory, physics (especially statistical mechanics), and computer science (especially algorithmic proof theory). Presented in a coherant and clear way, this will be the first book entirely devoted to the Rogers—Ramanujan identities and will include related historical material that is unavailable elsewhere.

An Invitation to Q-series

An Invitation to Q-series
Author: Hei-Chi Chan
Publisher: World Scientific
Total Pages: 237
Release: 2011
Genre: Mathematics
ISBN: 9814343846

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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.

Invitation To Q-series, An: From Jacobi's Triple Product Identity To Ramanujan's "Most Beautiful Identity"

Invitation To Q-series, An: From Jacobi's Triple Product Identity To Ramanujan's
Author: Hei-chi Chan
Publisher: World Scientific
Total Pages: 237
Release: 2011-04-04
Genre: Mathematics
ISBN: 9814460583

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The aim of these 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.

An Introduction to q-analysis

An Introduction to q-analysis
Author: Warren P. Johnson
Publisher: American Mathematical Soc.
Total Pages: 519
Release: 2020-10-06
Genre: Education
ISBN: 1470456230

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Starting from simple generalizations of factorials and binomial coefficients, this book gives a friendly and accessible introduction to q q-analysis, a subject consisting primarily of identities between certain kinds of series and products. Many applications of these identities to combinatorics and number theory are developed in detail. There are numerous exercises to help students appreciate the beauty and power of the ideas, and the history of the subject is kept consistently in view. The book has few prerequisites beyond calculus. It is well suited to a capstone course, or for self-study in combinatorics or classical analysis. Ph.D. students and research mathematicians will also find it useful as a reference.

An Invitation to Q-series

An Invitation to Q-series
Author: Chan Hei-Chi
Publisher:
Total Pages: 226
Release: 2011
Genre: q-series
ISBN:

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Ramanujan's Place in the World of Mathematics

Ramanujan's Place in the World of Mathematics
Author: Krishnaswami Alladi
Publisher: Springer Nature
Total Pages: 265
Release: 2021-09-17
Genre: Mathematics
ISBN: 9811562415

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The First Edition of the book is a collection of articles, all by the author, on the Indian mathematical genius Srinivasa Ramanujan as well as on some of the greatest mathematicians in history whose life and works have things in common with Ramanujan. It presents a unique comparative study of Ramanujan’s spectacular discoveries and remarkable life with the monumental contributions of various mathematical luminaries, some of whom, like Ramanujan, overcame great difficulties in life. Also, among the articles are reviews of three important books on Ramanujan’s mathematics and life. In addition, some aspects of Ramanujan’s contributions, such as his remarkable formulae for the number pi, his path-breaking work in the theory of partitions, and his fundamental observations on quadratic forms, are discussed. Finally, the book describes various current efforts to ensure that the legacy of Ramanujan will be preserved and continue to thrive in the future. This Second Edition is an expanded version of the first with six more articles by the author. Of note is the inclusion of a detailed review of the movie The Man Who Knew Infinity, a description of the fundamental work of the SASTRA Ramanujan Prize Winners, and an account of the Royal Society Conference to honour Ramanujan’s legacy on the centenary of his election as FRS.

$q$-Series with Applications to Combinatorics, Number Theory, and Physics

$q$-Series with Applications to Combinatorics, Number Theory, and Physics
Author: Bruce C. Berndt
Publisher: American Mathematical Soc.
Total Pages: 290
Release: 2001
Genre: Mathematics
ISBN: 0821827464

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The subject of $q$-series can be said to begin with Euler and his pentagonal number theorem. In fact, $q$-series are sometimes called Eulerian series. Contributions were made by Gauss, Jacobi, and Cauchy, but the first attempt at a systematic development, especially from the point of view of studying series with the products in the summands, was made by E. Heine in 1847. In the latter part of the nineteenth and in the early part of the twentieth centuries, two Englishmathematicians, L. J. Rogers and F. H. Jackson, made fundamental contributions. In 1940, G. H. Hardy described what we now call Ramanujan's famous $ 1\psi 1$ summation theorem as ``a remarkable formula with many parameters.'' This is now one of the fundamental theorems of the subject. Despite humble beginnings,the subject of $q$-series has flourished in the past three decades, particularly with its applications to combinatorics, number theory, and physics. During the year 2000, the University of Illinois embraced The Millennial Year in Number Theory. One of the events that year was the conference $q$-Series with Applications to Combinatorics, Number Theory, and Physics. This event gathered mathematicians from the world over to lecture and discuss their research. This volume presents nineteen of thepapers presented at the conference. The excellent lectures that are included chart pathways into the future and survey the numerous applications of $q$-series to combinatorics, number theory, and physics.

Special Functions, $q$-Series and Related Topics

Special Functions, $q$-Series and Related Topics
Author: Mourad Ismail
Publisher: American Mathematical Soc.
Total Pages: 289
Release: 1997
Genre: Mathematics
ISBN: 082180524X

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This book contains contributions from the proceedings at The Fields Institute workshop on Special Functions, q-Series and Related Topics that was held in June 1995. The articles cover areas from quantum groups and their representations, multivariate special functions, q-series, and symbolic algebra techniques as well as the traditional areas of single-variable special functions. The book contains both pure and applied topics and reflects recent trends of research in the various areas of special functions.