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Power of q: A Personal Journey Softcover reprint of the original 1st ed. 2017 [Minkštas viršelis]

  • Formatas: Paperback / softback, 415 pages, aukštis x plotis: 235x155 mm, weight: 6672 g, XXIV, 415 p., 1 Paperback / softback
  • Serija: Developments in Mathematics 49
  • Išleidimo metai: 04-Aug-2018
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319862413
  • ISBN-13: 9783319862415
Kitos knygos pagal šią temą:
Power of q: A Personal Journey Softcover reprint of the original 1st ed. 2017Didesnis paveikslėlis
  • Formatas: Paperback / softback, 415 pages, aukštis x plotis: 235x155 mm, weight: 6672 g, XXIV, 415 p., 1 Paperback / softback
  • Serija: Developments in Mathematics 49
  • Išleidimo metai: 04-Aug-2018
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319862413
  • ISBN-13: 9783319862415
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319862415.html

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.

Recenzijos

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 authors own papers. those who already love q-series will find much to admire and enjoy in Hirschhorns 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)

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
RogersRamanujan identities and the RogersRamanujan continued fraction.-
16. The series expansion of the RogersRamanujan continued fraction.- 17. The
2- and 4-dissections of Ramanujans continued fraction and its reciprocal.-
18. The series expansion of the Ramanujan-Gollnitz-Gordon continued fraction
and its reciprocal.- 19. Jacobis 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
JordanKronecker identity.- 29. Melhams 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.