**Mathematical Surveys and Monographs**

Volume: 37;
1991;
412 pp;
Hardcover

MSC: Primary 19; 30; 33; 39;
Secondary 11; 51; 57

Print ISBN: 978-0-8218-1634-9

Product Code: SURV/37

List Price: $154.00

Individual Member Price: $123.20

**Electronic ISBN: 978-1-4704-1264-7
Product Code: SURV/37.E**

List Price: $154.00

Individual Member Price: $123.20

# Structural Properties of Polylogarithms

Share this page *Edited by *
*Leonard Lewin*

Years ago, the handful of peculiar numerical dilogarithmic identities, known since the time of Euler and Landen, gave rise to new discoveries concerning cyclotomic equations and related polylogarithmic ladders. These discoveries were made mostly by the methods of classical analysis, with help from machine computation. About the same time, starting with Bloch's studies on the application of the dilogarithm in algebraic \(K\)-theory and algebraic geometry, many important discoveries were made in diverse areas.

This book seeks to provide a synthesis of these two streams of thought. In addition to an account of ladders and their association with functional equations, the chapters include applications to volume calculations in Lobatchevsky geometry, relations to partition theory, connections with Clausen's function, new functional equations, and applications to \(K\)-theory and other branches of abstract algebra. This rapidly-expanding field is brought up to date with two appendices, and the book concludes with an extensive bibliography of recent publications. About two-thirds of the material is accessible to mathematicians and scientists in many areas, while the remainder requires more specialized background in abstract algebra.

#### Table of Contents

# Table of Contents

## Structural Properties of Polylogarithms

- Contents vii8 free
- Preface xiii14 free
- Acknowledgments xv16 free
- List of Contributors xvii18 free
- Chapter 1. The Evolution of the Ladder Concept 120 free
- Chapter 2. Dilogarithmic Ladders 1130
- 2.1 Derivation from Kummer's Functional Equation 1130
- 2.2 Relation to Clausen's Function 1534
- 2.3 A Three-Variable Dilogarithmic Functional Equation 1736
- 2.4 Functional Equations in the Complex Plane 1837
- 2.5 Cyclotomic Equations and Rogers' Function 2039
- 2.6 Accessible and Analytic Ladders 2140
- 2.7 Inaccessible Ladders 2342
- References 2544

- Chapter 3. Polylogarithmic Ladders 2746
- 3.1 Kummer's Function and its Relation to the Polylogarithm 2746
- 3.2 Functional Equations for the Polylogarithm 2847
- 3.3 A Generalization of Rogers' Function to the nth Order 3150
- 3.4 Ladder Order-Independence on Reduction of Order 3352
- 3.5 Generic Ladders for the Base Equation u[sup(p)] + u[sup(q)] = 1 3453
- 3.6 Examples of Ladders for n ≤ 3 4059
- 3.7 Examples of Ladders for n ≤ 4 4463
- 3.8 Examples of Ladders for n ≤ 5 4564
- 3.9 Polynomial Relations for Ladders 4665
- References 4766

- Chapter 4. Ladders in the Trans-Kummer Region 4968
- Chapter 5. Supemumary Ladders 6988
- 5.1 The Concept of Supemumary Results 6988
- 5.2 Supemumary Results for p = 4 7190
- 5.3 Supemumary Results for p = 5 7695
- 5.4 Supemumary Results for p = 6 7897
- 5.5 Supemumary Results for the Equation-family u[sup(6m+1)] + u[sup(6r-1)] = 1 8099
- 5.6 Supemumary Results for an Irreducible Quintic 82101
- 5.7 Supemumary Ladders from a 15-Term Functional Equation 84103
- 5.8 Supemumary Ladders on the Unit Circle 90109
- References 96115

- Chapter 6. Functional Equations and Ladders 97116
- Chapter 7. Multivariable Polylogarithm Identities 123142
- 7.0 Introduction 123142
- 7.1 A General Identity for the Dilogarithm 123142
- 7.2 A General Identity for the Bloch-Wigner Function 135154
- 7.3 A General Identity for the Trilogarithm and D[sub(3)](z) 141160
- 7.4 Linear Power Relations among Dilogarithms 147166
- 7.5 Cyclotomic Equations and Bases for Polylogarithm Relations 154173
- 7.6 Mahler's Measure and Salem/Pisot Numbers 160179
- 7.7 Recent Results for Supemumary Ladders 165184
- References 168187

- Chapter 8. Functional Equations of Hyperlogarithms 171190
- Chapter 9. Kummer-Type Functional Equations of Polylogarithms 185204
- Chapter 10. The Basic Structure of Polylogarithmic Equations 205224
- 10.1 Introduction 205224
- 10.2 Canonical Unipotent Connection on P[sup(1)](C)\{a[sub(1)],...,a[sub(n+1)]} 211230
- 10.3 Horizontal Sections 213232
- 10.4 Easy Lemmas about Monodromy 215234
- 10.5 Functional Equations 216235
- 10.6 Functional Equations of Polylogarithms 218237
- 10.7 Functional Equations of Lower Degree Polylogarithms 223242
- 10.8 Generalized Bloch Groups 228247
- Acknowledgements 231250
- References 231250

- Chapter 11. K-Theory, Cyclotomic Equations and Clausen's Function 233252
- Chapter 12. Function Theory of Polylogarithms 275294
- Chapter 13. Partition Identities and the Dilogarithm 287306
- Chapter 14. The Dilogarithm and Volumes of Hyperbolic Polytopes 301320
- Chapter 15. Introduction to Higher Logarithms 337356
- 15.1 The Problem of Generalizing the Logarithm and the Dilogarithm 337356
- 15.2 The Quest for Higher Logarithms 340359
- 15.3 Higher Logarithms 341360
- 15.4 The Higher Logarithm Bicomplex 343362
- 15.5 Multivalued Deligne Cohomology 346365
- 15.6 Higher Logarithms as Deligne Cohomology Classes 350369
- Acknowledgements 351370
- References 352371

- Chapter 16. Some Miscellaneous Results 355374
- Appendix A. Special Values and Functional Equations of Polylogarithms 377396
- 0. Introduction 377396
- 1. The Basic Algebraic Relation and the Definition of A[sub(m)](F) 378397
- 2. Examples of Dilogarithm Relations 383402
- 3. Examples for Higher Order Polylogarithms 385404
- 4. Examples: Ladders 387406
- 5. Existence of Relations among Polylogarithm Values of Arbitrarily High Order 390409
- 6. A Conjecture on Linear Independence 391410
- 7. Functional Equations 392411
- References 399418

- Appendix B. Summary of the Informal Polylogarithm Workshop, 401420
- Bibliography 405424
- Index 409428

#### Reviews

A remarkable book on a remarkable function, brimming with the enthusiasm of new discoveries and ideas for further research.

-- Zentralblatt MATH