Coverart for item
The Resource The Power of q : A Personal Journey, by Michael D. Hirschhorn, (electronic resource)

The Power of q : A Personal Journey, by Michael D. Hirschhorn, (electronic resource)

Label
The Power of q : A Personal Journey
Title
The Power of q
Title remainder
A Personal Journey
Statement of responsibility
by Michael D. Hirschhorn
Creator
Subject
Language
eng
Summary
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.--
Member of
Assigning source
Provided by publisher
http://library.link/vocab/creatorName
Hirschhorn, Michael D
Image bit depth
0
Literary form
non fiction
Nature of contents
dictionaries
Series statement
  • Springer eBooks
  • Developments in Mathematics,
Series volume
49
http://library.link/vocab/subjectName
  • Mathematics
  • Number theory
Label
The Power of q : A Personal Journey, by Michael D. Hirschhorn, (electronic resource)
Link
https://eui.idm.oclc.org/login?url=http://dx.doi.org/10.1007/978-3-319-57762-3
Instantiates
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
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. Some modular 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
Control code
978-3-319-57762-3
Dimensions
unknown
Extent
1 online resource (XXII, 415 pages)
File format
multiple file formats
Form of item
  • online
  • electronic
Governing access note
Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, non-commercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
Isbn
9783319577623
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-319-57762-3
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1000451420
Label
The Power of q : A Personal Journey, by Michael D. Hirschhorn, (electronic resource)
Link
https://eui.idm.oclc.org/login?url=http://dx.doi.org/10.1007/978-3-319-57762-3
Publication
Antecedent source
mixed
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Color
not applicable
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
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. Some modular 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
Control code
978-3-319-57762-3
Dimensions
unknown
Extent
1 online resource (XXII, 415 pages)
File format
multiple file formats
Form of item
  • online
  • electronic
Governing access note
Use of this electronic resource may be governed by a license agreement which restricts use to the European University Institute community. Each user is responsible for limiting use to individual, non-commercial purposes, without systematically downloading, distributing, or retaining substantial portions of information, provided that all copyright and other proprietary notices contained on the materials are retained. The use of software, including scripts, agents, or robots, is generally prohibited and may result in the loss of access to these resources for the entire European University Institute community
Isbn
9783319577623
Level of compression
uncompressed
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-319-57762-3
Quality assurance targets
absent
Reformatting quality
access
Specific material designation
remote
System control number
(OCoLC)1000451420

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