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An Introduction to Q-Analysis An Introductionq-analysis to Warren P. Johnson 10.1090/mbk/134 An Introduction to q-analysis An Introduction to q-analysis Warren P. Johnson 2010 Mathematics Subject Classification. Primary 05A30, 05A17, 11P84, 11P81, 33D15. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-134 Library of Congress Cataloging-in-Publication Data Names: Johnson, Warren Pierstorff, 1960- author. Title: Introduction to q-analysis / Warren P. Johnson. Description: Providence : American Mathematical Society, 2020. | Includes bibliographical refer- ences and index. | Summary: Identifiers: LCCN 2020021142 | ISBN 9781470456238 (paperback) | ISBN 9781470462109 (ebook) Subjects: LCSH: Combinatorial analysis. | Graph theory. | Number theory. | AMS: Combinatorics – Enumerative combinatorics – q-calculus and related topics. | Combinatorics – Enumerative combinatorics – Partitions of integers. | Number theory – Additive number theory; partitions – Partition identities; identities of Rogers-Ramanujan type. | Number theory – Additive number theory; partitions – Elementary theory of partitions. | Special functions. Classification: LCC QA164 .J54 2020 — DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2020021142 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2020 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 252423222120 Contents An Introduction to q-analysis xi Chapter 1. Inversions 1 1.1. Stern’s problem 1 Exercises 5 1.2. The q-factorial 7 Exercises 11 1.3. q-binomial coefficients 14 Exercises 19 1.4. Some identities for q-binomial coefficients 20 Exercises 23 1.5. Another property of q-binomial coefficients 25 Exercises 27 1.6. q-multinomial coefficients 29 Exercises 31 1.7. The Z-identity 33 Exercises 36 1.8. Bibliographical Notes 37 Chapter 2. q-binomial Theorems 39 2.1. A noncommutative q-binomial Theorem 39 Exercises 43 2.2. Potter’s proof 45 Exercises 47 2.3. Rothe’s q-binomial theorem 49 Exercises 53 2.4. The q-derivative 57 Exercises 59 2.5. Two q-binomial theorems of Gauss 61 Exercises 66 2.6. Jacobi’s q-binomial theorem 71 Exercises 72 2.7. MacMahon’s q-binomial theorem 74 Exercises 77 2.8. A partial fraction decomposition 79 Exercises 82 2.9. A curious q-identity of Euler, and some extensions 82 Exercises 86 2.10. The Chen–Chu–Gu identity 88 Exercises 91 v vi CONTENTS 2.11. Bibliographical Notes 91 Chapter 3. Partitions I: Elementary Theory 93 3.1. Partitions with distinct parts 93 Exercises 95 3.2. Partitions with repeated parts 98 Exercises 103 3.3. Ferrers diagrams 106 Exercises 113 3.4. q-binomial coefficients and partitions 116 Exercises 119 3.5. An identity of Euler, and its “finite” form 120 Exercises 126 3.6. Another identity of Euler, and its finite form 128 Exercises 130 3.7. The Cauchy/Crelle q-binomial series 132 Exercises 137 3.8. q-exponential functions 141 Exercises 145 3.9. Bibliographical Notes 148 Chapter 4. Partitions II: Geometric Theory 149 4.1. Euler’s pentagonal number theorem 149 Exercises 153 4.2. Durfee squares 157 Exercises 162 4.3. Euler’s pentagonal number theorem: Franklin’s proof 164 Exercises 167 4.4. Divisor sums 167 Exercises 173 4.5. Sylvester’s fishhook bijection 180 Exercises 187 4.6. Bibliographical Notes 188 Chapter 5. More q-identities: Jacobi, Gauss, and Heine 191 5.1. Jacobi’s triple product 191 Exercises 195 5.2. Other proofs and related results 201 Exercises 205 5.3. The quintuple product identity 214 Exercises 218 5.4. Lebesgue’s identity 221 Exercises 223 5.5. Basic hypergeometric series 227 Exercises 230 5.6. More 2φ1 identities 233 Exercises 236 5.7. The q-Pfaff–Saalsch¨utz identity 239 Exercises 241 CONTENTS vii 5.8. Bibliographical Notes 243 Chapter 6. Ramanujan’s 1ψ1 Summation Formula 247 6.1. Ramanujan’s formula 247 Exercises 249 6.2. Four proofs 250 Exercises 253 6.3. From the q-Pfaff–Saalsch¨utz sum to Ramanujan’s 1ψ1 summation 256 Exercises 259 6.4. Another identity of Cauchy, and its finite form 259 Exercises 260 6.5. Cauchy’s “mistaken identity” 263 Exercises 265 6.6. Ramanujan’s formula again 266 Exercises 268 6.7. Bibliographical Notes 268 Chapter 7. Sums of Squares 271 7.1. Cauchy’s formula 271 Exercises 272 7.2. Sums of two squares 276 Exercises 278 7.3. Sums of four squares 281 Exercises 286 7.4. Bibliographical Notes 288 Chapter 8. Ramanujan’s Congruences 289 8.1. Ramanujan’s congruences 289 Exercises 291 8.2. Ramanujan’s “most beautiful” identity 292 Exercises 298 8.3. Ramanujan’s congruences again 300 8.4. Bibliographical Notes 303 Chapter 9. Some Combinatorial Results 305 9.1. Revisiting the q-factorial 305 Exercises 309 9.2. Revisiting the q-binomial coefficients 311 Exercises 314 9.3. Foata’s bijection for q-multinomial coefficients 316 Exercises 319 9.4. MacMahon’s proof 319 Exercises 321 9.5. q-derangement numbers 323 Exercises 329 9.6. q-Eulerian numbers and polynomials 331 Exercises 338 9.7. q-trigonometric functions 338 Exercises 342 9.8. Combinatorics of q-tangents and secants 343 viii CONTENTS 9.9. Bibliographical Notes 349 Chapter 10. The Rogers–Ramanujan Identities I: Schur 351 10.1. Schur’s extension of Franklin’s argument 351 Exercises 356 10.2. The Bressoud–Chapman proof 357 Exercises 361 10.3. The AKP and GIS identities 363 10.4. Schur’s second partition theorem 365 Exercises 370 10.5. Bibliographical Notes 375 Chapter 11. The Rogers–Ramanujan Identities II: Rogers 377 11.1. Ramanujan’s proof 377 Exercises 381 11.2. The Rogers–Ramanujan identities and partitions 383 Exercises 388 11.3. Rogers’s second proof 388 Exercises 391 11.4. More identities of Rogers 394 Exercises 399 11.5. Rogers’s identities and partitions 399 11.6. The G¨ollnitz–Gordon identities 403 Exercises 407 11.7. The G¨ollnitz–Gordon identities and partitions 412 Exercises 414 11.8. Bibliographical Notes 416 Chapter 12. The Rogers–Selberg Function 417 12.1. The Rogers–Selberg function 417 Exercises 419 12.2. Some applications 420 Exercises 423 12.3. The Selberg coefficients 423 Exercises 427 12.4. The case k = 3 427 12.5. Explicit formulas for the Q functions 429 Exercises 430 12.6. Explicit formulas for S3,i(x) 430 Exercises 431 12.7. The payoff for k = 3 432 Exercises 434 12.8. Gordon’s theorem 434 12.9. Bibliographical Notes 436 Chapter 13. Bailey’s 6ψ6 Sum 437 13.1. Bailey’s formula 437 Exercises 439 13.2. Another proof of Ramanujan’s “most beautiful” identity 442 13.3. Sums of eight squares and of eight triangular numbers 444 CONTENTS ix Exercises 447 13.4. Bailey’s 6ψ6 summation formula 449 Exercises 450 13.5. Askey’s proof: Phase 1 454 Exercises 457 13.6. Askey’s proof: Phase 2 457 Exercises 460 13.7. Askey’s proof: Phase 3 460 Exercises 461 13.8. An integral 465 Exercises 470 13.9. Bailey’s lemma 471 13.10. Watson’s transformation 475 Exercises 479 13.11. Bibliographical Notes 481 Appendix A. A Brief Guide to Notation 483 Appendix B. Infinite Products 487 Exercises 491 Appendix C. Tannery’s Theorem 495 Bibliography 501 Index of Names 513 Index of Topics 517 An Introduction to q-analysis The phrase “q-analysis” was used in the first referee’s report I ever got. While the subject of this book has a flavor all its own, and has been studied for almost 300 years, there is no term in common use that describes it really well. The closest standard name is “q-series”, which is not bad—finite and infinite series occur almost everywhere, as does the letter q—but it is a little too restrictive. I think it needs another appellation, q-analysis is the best one I can think of, and I thank that anonymous referee for it (and an excellent report). Peter Paule used it in [181]. I have tried very hard to write a book that can be read by undergraduates. The prerequisites are minimal. One cannot have “the fear of all sums” that plagues many calculus students, but very little specific knowledge of calculus 2 will be required. In particular, you do not need an extensive knowledge of convergence tests, since for q-series the ratio test is nearly always appropriate. (The root test is marginally better in a few cases, and once in a while the nth term test is helpful.) Moreover, we will be much less concerned with when or whether an infinite series convergesthanwithwhatitconvergesto. We will also be seeing zillions of finite and infinite products of a certain kind (this is one reason why I don’t want to just say “q-series”), but no prior knowledge of these is assumed.
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