**Mathematical Surveys and Monographs**

Volume: 27;
1988;
124 pp;
Softcover

MSC: Primary 33;
Secondary 05; 11

**Print ISBN: 978-0-8218-1524-3
Product Code: SURV/27**

List Price: $41.00

AMS Member Price: $32.80

MAA Member Price: $36.90

**Electronic ISBN: 978-1-4704-1254-8
Product Code: SURV/27.E**

List Price: $38.00

AMS Member Price: $30.40

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# Basic Hypergeometric Series and Applications

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*Nathan Fine*

The theory of partitions, founded by Euler, has led in
a natural way to the idea of basic hypergeometric series, also known
as Eulerian series. These series were first studied systematically
by Heine, but many early results are attributed to Euler, Gauss,
and Jacobi. Today, research in \(q\)-hypergeometric series is
very active, and there are now major interactions with Lie
algebras, combinatorics, special functions, and number theory.

However, the theory has been developed to such an extent and
with such a profusion of powerful and general results that the subject
can appear quite formidable to the uninitiated. By providing a
simple approach to basic hypergeometric series, this book provides
an excellent elementary introduction to the subject.

The starting point is a simple function of several
variables satisfying a number of \(q\)-difference equations. The
author presents an elementary method for using these equations to
obtain transformations of the original function. A bilateral series,
formed from this function, is summed as an infinite product,
thereby providing an elegant and fruitful result which goes back to
Ramanujan. By exploiting a special case, the author is able to evaluate
the coefficients of several classes of infinite products in terms
of divisor sums. He also touches on general transformation theory
for basic series in many variables and the basic multinomial, which is
a generalization of a finite sum.

These developments lead naturally to the arithmetic domains
of partition theory, theorems of Liouville type, and sums of
squares. Contact is also made with the mock theta-functions of
Ramanujan, which are linked to the rank of partitions. The author gives
a number of examples of modular functions with multiplicative
coefficients, along with the beginnings of an elementary constructive
approach to the field of modular equations.

Requiring only an undergraduate background in mathematics, this book
provides a rapid entry into the field. Students of partitions, basic
series, theta-functions, and modular equations, as well as
research mathematicians interested in an elementary approach to these
areas, will find this book useful and enlightening. Because of the
simplicity of its approach and its accessibility, this work may prove
useful as a textbook.

#### Reviews & Endorsements

Rich with examples and results from the theory of partitions, the study of Ramanujan's mock theta functions, and modular equations.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Basic Hypergeometric Series and Applications

- Contents vii8 free
- Foreword ix10 free
- Introduction xi12 free
- Chapter 1. Fundamental Properties of Basic Hypergeometric Series 118 free
- 1. Definitions 118
- 2. Two functional equations 118
- 3. The analytic character of F(a,b;t) 219
- 4. More transformations 219
- 5. The method of iteration 320
- 6. Application of iteration (t → tq) 421
- 7. Iteration of b → bq 522
- 8. Consequences of §7 724
- 9. Further consequences of §7 825
- 10. A product-series identity 1027
- 11. Iteration of a → aq 1229
- 12. Iteration of (b,t) → (bq,tq) 1330
- 13. Iteration of (a,t) → (aq,tq) 1431
- 14. Iteration of (a,b,t) → (aq,bq,tq) 1532
- 15. A special development 1633
- 16. The partial-fraction decomposition 1734
- 17. Jacobi's triple product 1835
- 18. A bilateral series 1936
- 19. Two product-series identities 2239
- 20. A general transformation 2441
- 21. The basic multinomial 3047
- Notes 3249
- References 3552

- Chapter 2. Partitions 3754
- Chapter 3. Mock Theta-Functions and the Functions L(N), J(N) 5572
- Chapter 4. Other Applications 6784
- Chapter 5. Modular Equations 93110
- 34. Modular equations, preliminaries 93110
- 35. A set of functional equations 96113
- 36. Application of (35.13) 97114
- 37. Two modular equations 100117
- 38. Continuation of §36 101118
- 39. Other functional values of H(z) 104121
- 40. A system of identities 107124
- 41. Permanent identities 112129
- 42. Continuation 117134
- Notes 121138
- References 122139

- Bibliography 123140