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Table of contents

  1. Front Matter
    Pages i-xxiv
  2. Michael D. Hirschhorn
    Pages 1-17
  3. Michael D. Hirschhorn
    Pages 19-26
  4. Michael D. Hirschhorn
    Pages 27-42
  5. Michael D. Hirschhorn
    Pages 43-54
  6. Michael D. Hirschhorn
    Pages 55-58
  7. Michael D. Hirschhorn
    Pages 59-70
  8. Michael D. Hirschhorn
    Pages 71-83
  9. Michael D. Hirschhorn
    Pages 85-92
  10. Michael D. Hirschhorn
    Pages 93-98
  11. Michael D. Hirschhorn
    Pages 99-108
  12. Michael D. Hirschhorn
    Pages 109-112
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    Pages 113-121
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    Pages 131-138
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    Pages 169-174
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    Pages 175-178
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    Pages 179-184
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    Pages 205-210
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    Pages 211-215
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    Pages 217-224
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    Pages 225-227
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    Pages 229-233
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    Pages 235-246
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    Pages 247-255
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    Pages 257-287
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    Pages 289-295
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    Pages 297-301
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    Pages 303-309
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    Pages 311-333
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    Pages 335-338
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    Pages 339-344
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    Pages 345-349
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    Pages 351-356
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    Pages 357-364
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    Pages 365-371
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  37. Michael D. Hirschhorn
    Pages 393-400
  38. Michael D. Hirschhorn
    Pages E1-E3
  39. Back Matter
    Pages 401-415

About this book

Introduction

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.

Keywords

q-series Ramanujan partition congruences dissection Euler product hypergeometric series quintuple product identity Winquist's identity Jacobi identity crank of partition forty identities continued fraction four-squares theorem two-squares theorem tau function Ramanujan partition congruences theory of representations supplementary text

Authors and affiliations

  • Michael D. Hirschhorn
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-57762-3
  • Copyright Information Springer International Publishing AG 2017
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-57761-6
  • Online ISBN 978-3-319-57762-3
  • Series Print ISSN 1389-2177
  • Series Online ISSN 2197-795X
  • Buy this book on publisher's site