Developments in Mathematics

The Power of q

A Personal Journey

Authors: Hirschhorn, Michael D.

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  • 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
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eBook 91,62 €
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  • ISBN 978-3-319-57762-3
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  • Immediate eBook download after purchase
Hardcover 114,39 €
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  • ISBN 978-3-319-57761-6
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  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 114,39 €
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  • ISBN 978-3-319-86241-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
About this book

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.

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)


Table of contents (44 chapters)

Table of contents (44 chapters)

Buy this book

eBook 91,62 €
price for Spain (gross)
  • ISBN 978-3-319-57762-3
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 114,39 €
price for Spain (gross)
  • ISBN 978-3-319-57761-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
Softcover 114,39 €
price for Spain (gross)
  • ISBN 978-3-319-86241-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
The Power of q
Book Subtitle
A Personal Journey
Authors
Series Title
Developments in Mathematics
Series Volume
49
Copyright
2017
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing AG
eBook ISBN
978-3-319-57762-3
DOI
10.1007/978-3-319-57762-3
Hardcover ISBN
978-3-319-57761-6
Softcover ISBN
978-3-319-86241-5
Series ISSN
1389-2177
Edition Number
1
Number of Pages
XXIV, 415
Topics