The Power of Q: A Personal Journey by Michael D Hirschhorn: New

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Book Title
The Power of Q: A Personal Journey
Publication Date
2017-08-16
Pages
415
ISBN
9783319577616
Kategorie

Über dieses Produkt

Product Identifiers

Publisher
Springer International Publishing A&G
ISBN-10
3319577611
ISBN-13
9783319577616
eBay Product ID (ePID)
236918950

Product Key Features

Number of Pages
Xxiv, 415 Pages
Publication Name
Power of Q : a Personal Journey
Language
English
Subject
Number Theory, Mathematical Analysis
Publication Year
2017
Type
Textbook
Author
Michael D. Hirschhorn
Subject Area
Mathematics
Series
Developments in Mathematics Ser.
Format
Hardcover

Dimensions

Item Weight
275.3 Oz
Item Length
9.3 in
Item Width
6.1 in

Additional Product Features

Reviews
"This book provides an introduction to q-series that would be accessible to calculus students, its main purpose is to offer beautiful theorems to the reader along with, in many instances, equally beautiful proofs that cannot be found elsewhere, except possibly in the author's own papers. ... those who already love q-series will find much to admire and enjoy in Hirschhorn's book The Power of q. Those desiring an introduction to the subject can also enjoy it." (Bruce Berndt, The American Mathematical Monthly, Vol. 126 (2), April, 2019)
Dewey Edition
23
TitleLeading
The
Series Volume Number
49
Number of Volumes
1 vol.
Illustrated
Yes
Dewey Decimal
515.243
Table Of Content
Foreword.- Preface.- 1. Introduction.- 2. Jacobi's two-squares and four-squares theorems.- 3. Ramanujan's partition congruences.- 4. Ramanujan's partition congruences-- a uniform proof.- 5. Ramanujan's "most beautiful identity".- 6. Ramanujan's partition congruences for powers of 5.- 7. Ramanujan's partition congruences for powers of 7.- 8. Ramanujan's 5-dissection of Euler's product.- 9. A "difficult and deep" identity of Ramanujan.- 10. The quintuple product identity.- 11. Winquist's identity.- 12. The crank of a partition.- 13. Two more proofs of p(11n + 6) ≡ 0 (mod 11), and more.- 14. Partitions where even parts come in two colours.- 15. The Rogers-Ramanujan identities and the Rogers-Ramanujan continued fraction.- 16. The series expansion of the Rogers-Ramanujan continued fraction.- 17. The 2- and 4-dissections of Ramanujan's continued fraction and its reciprocal.- 18. The series expansion of the Ramanujan-Gollnitz-Gordon continued fraction and its reciprocal.- 19. Jacobi's " aequatio identica satis abstrusa ".- 20. Two modular equations.- 21. A letter from Fitzroy House.- 22. The cubic functions of Borwein, Borwein and Garvan.- 23. Some classical results on representations.- 24. Further classical results on representations.- 25. Further results on representations.- 26. Even more representation results.- 27. Representation results and Lambert series.- 28. The Jordan-Kronecker identity.- 29. Melham's identities.- 30. Partitions into four squares.- 31. Partitions into four distinct squares of equal parity.- 32. Partitions with odd parts distinct.- 33. Partitions with even parts distinct.- 34. Some identities involving phi( q ) and psi( q ).- 35. Some useful parametrisations.- 36. Overpartitions.- 37. Bipartitions with odd parts distinct.- 38. Overcubic partitions.- 39. Generalised Frobenius partitions.- 40. Somemodular equations of Ramanujan.- 41. Identities involving k = qR(q)R(q2)2 .- 42. Identities involving v=q1/2(q,q7;q8)infinity/(q3,q5;q8)infinity .- 43. Ramanujan's tau function.- Appendix.- Index.
Synopsis
Aptly conveys the beauty and power of q-series Accessible to advanced undergraduates, graduate students, and researchers Historical notes enrich the readers understanding of the subject First monograph to focus uniquely on q-series, 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 Erdos couldn't do." The book concludes with a look at Ramanujan's remarkable tau function.
LC Classification Number
QA241-247.5

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