Contents

XI

§11. Some special cases (restricted characteristics) 260

11.1. Characteristics with m! = k 260

11.2. Characteristics with m = k 260

11.3. Ratios 263

Chapter 4. Theta constant identities 265

§1. Dimension considerations 268

1.1. The septuple product identity 268

1.2. Further generalizations 272

§2. Uniformization considerations 274

§3. Elliptic functions as quotients of iV-th order theta functions 274

3.1. The Jacobi quartic and derivative formula revisited 274

3.2. More identities - revisited 276

3.3. More identities - new results 277

3.4. More first order applications 279

3.5. Some modular equations 284

§4. Identities which arise from modular forms 291

4.1. Multiplicative meromorphic forms 292

4.2. Cusp forms for T 294

4.3. Some special results for the primes 5 and 7 297

§5. Ramanujan's r-function 297

§6. Identities among infinite products 299

§7. Identities via logarithmic differentiation 301

§8. Averaging automorphic forms 308

§9. The groups G(k) 312

Chapter 5. Partition theory: Ramanujan congruences 325

§1. Calculations of PN(^) 331

§2. Some preliminaries 333

2.1. T (p, g)-invariant functions 333

2.2. Calculation of divisor of 7?(iV) 342

2.3. Coset representatives 344

§3. Generalities on constructions of ro(fc)-invariant functions 345

3.1. The basic problems 345