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Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

2007 Mathematics
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Release : 2007
Genre : Mathematics
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Book Rating : 840/5 ( reviews)

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Book Synopsis Ramanujan's Forty Identities for the Rogers-Ramanujan Functions by : Bruce C. Berndt

Download or read book Ramanujan's Forty Identities for the Rogers-Ramanujan Functions written by Bruce C. Berndt. This book was released on 2007. Available in PDF, EPUB and Kindle. Book excerpt: Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, ""Watson, look up at the stars and tell me what you deduce."" Watson says, ""I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets."" Holmes replies, ""Watson, you idiot. Somebody stole our tent."" When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an ""idiot."" He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities. Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. In particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought ""more natural"" proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for the Rogers-Ramanujan functions. In this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several identities, the authors present two or three proofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan's lost notebook, is what Ramanujan might have used to discover several of the more difficult identities. Some of the new methods in this paper can be employed to establish new identities for the Rogers-Ramanujan functions.

Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

2007 Mathematics
Download Ramanujan's Forty Identities for the Rogers-Ramanujan Functions PDF Online Free

Author :
Release : 2007
Genre : Mathematics
Kind : eBook
Book Rating : 73X/5 ( reviews)

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Book Synopsis Ramanujan's Forty Identities for the Rogers-Ramanujan Functions by : Bruce C. Berndt

Download or read book Ramanujan's Forty Identities for the Rogers-Ramanujan Functions written by Bruce C. Berndt. This book was released on 2007. Available in PDF, EPUB and Kindle. Book excerpt: Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his sidekick Dr. Watson go camping and pitch their tent under the stars. During the night, Holmes wakes his companion and says, ``Watson, look up at the stars and tell me what you deduce.'' Watson says, ``I see millions of stars, and it is quite likely that a few of them are planets just like Earth. Therefore there may also be life on these planets.'' Holmes replies, ``Watson, you idiot. Somebody stole ourtent.'' When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan functions, Watson, i.e., G. N. Watson, was not an ``idiot.'' He, L. J. Rogers, and D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli devised proofs for most (but not all) of the remaining identities.Although some of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those found by Ramanujan, those of Biagioli are not. in particular, Biagioli used the theory of modular forms. Haunted by the fact that little progress has been made into Ramanujan's insights on these identities in the past 85 years, the present authors sought ``more natural'' proofs. Thus, instead of a missing tent, we have had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for theRogers-Ramanujan functions. in this paper, for 35 of the 40 identities, the authors offer proofs that are in the spirit of Ramanujan. Some of the proofs presented here are due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several identities, the authors present two or threeproofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan's lost notebook, is what Ramanujan might have used to discover several of the more difficult identities. Some of the new methods in this paper can be employed to establish new identities for the Rogers-Ramanujanfunctions.

Modular identities for the Rogers-Ramanujan functions and analogues

2011
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Release : 2011
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Book Synopsis Modular identities for the Rogers-Ramanujan functions and analogues by : Chadwick Gugg

Download or read book Modular identities for the Rogers-Ramanujan functions and analogues written by Chadwick Gugg. This book was released on 2011. Available in PDF, EPUB and Kindle. Book excerpt: In his notebooks, Ramanujan recorded 40 beautiful modular relations for the Rogers-Ramanujan functions. Of these 40 identities, precisely one involves powers of the Rogers-Ramanujan functions. Ramanujan added the enigmatic note that "Each of these formulae is the simplest of a large class." This suggests that there are further modular identities involving powers of the Rogers-Ramanujan functions. Although numerous authors have studied identities for the Rogers-Ramanujan functions and various analogues, no systematic study of identities involving powers of the Rogers-Ramanujan functions has been undertaken. In this thesis, we continue the study of modular identities for the Rogers-Ramanujan functions, with particular emphasis on relations involving powers of the Rogers-Ramanujan functions. Our methods are classical, using tools and techniques that Ramanujan could have employed. These tools include, for example, manipulation of infinite series and the theory of modular equations. It is hoped that these methods will give new insights into these equations, and perhaps lead to understanding or discovering further families of identities of mathematical interest. Identities involving squares, cubes, fourth, and fifth powers of the Rogers-Ramanujan functions are enunciated and proved; many of these relations are new. Rich applications are made to the study of modular relations for the Rogers-Ramanujan continued fraction. To demonstrate the generality of our methods, analogous results are obtained in various cases for the Gollnitz-Gordon functions and the Ramanujan-Gollnitz-Gordon continued fraction. Further identities for the Rogers-Ramanujan functions, of the types found in Ramanujan's list of 40 relations for the Rogers-Ramanujan functions, are also studied. Analogous identities are obtained for the Gollnitz-Gordon functions, as well as for dodecic and sextodecic analogues of the Rogers-Ramanujan functions.

Ramanujan's Lost Notebook

2012-06-08 Mathematics
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Release : 2012-06-08
Genre : Mathematics
Kind : eBook
Book Rating : 101/5 ( reviews)

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Book Synopsis Ramanujan's Lost Notebook by : George E. Andrews

Download or read book Ramanujan's Lost Notebook written by George E. Andrews. This book was released on 2012-06-08. Available in PDF, EPUB and Kindle. Book excerpt: In the spring of 1976, George Andrews of Pennsylvania State University visited the library at Trinity College, Cambridge to examine the papers of the late G.N. Watson. Among these papers, Andrews discovered a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan. This manuscript was soon designated, "Ramanujan's lost notebook." Its discovery has frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony. This volume is the third of five volumes that the authors plan to write on Ramanujan’s lost notebook and other manuscripts and fragments found in The Lost Notebook and Other Unpublished Papers, published by Narosa in 1988. The ordinary partition function p(n) is the focus of this third volume. In particular, ranks, cranks, and congruences for p(n) are in the spotlight. Other topics include the Ramanujan tau-function, the Rogers–Ramanujan functions, highly composite numbers, and sums of powers of theta functions. Review from the second volume: "Fans of Ramanujan's mathematics are sure to be delighted by this book. While some of the content is taken directly from published papers, most chapters contain new material and some previously published proofs have been improved. Many entries are just begging for further study and will undoubtedly be inspiring research for decades to come. The next installment in this series is eagerly awaited." - MathSciNet Review from the first volume: "Andrews a nd Berndt are to be congratulated on the job they are doing. This is the first step...on the way to an understanding of the work of the genius Ramanujan. It should act as an inspiration to future generations of mathematicians to tackle a job that will never be complete." - Gazette of the Australian Mathematical Society

Combinatorial Number Theory

2011-12-22 Mathematics
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Release : 2011-12-22
Genre : Mathematics
Kind : eBook
Book Rating : 095/5 ( reviews)

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Book Synopsis Combinatorial Number Theory by : Bruce Landman

Download or read book Combinatorial Number Theory written by Bruce Landman. This book was released on 2011-12-22. Available in PDF, EPUB and Kindle. Book excerpt: This carefully edited volume contains selected refereed papers based on lectures presented by many distinguished speakers at the "Integers Conference 2005", an international conference in combinatorial number theory. The conference was held in celebration of the 70th birthday of Ronald Graham, a leader in several fields of mathematics.

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