مقدمه‌ای بر تحلیل $q$

Starting from simple generalizations of factorials and binomial coefficients, this book gives a friendly and accessible introduction to $q$-analysis, a subject consisting primarily of identities between certain kinds of series and products. Many applications of these identities to combinatorics and number theory are developed in detail. There are numerous exercises to help students appreciate the beauty and power of the ideas, and the history of the subject is kept consistently in view. The book has few prerequisites beyond calculus. It is well suited to a capstone course, or for self-study in combinatorics or classical analysis. Ph.D. students and research mathematicians will also find it useful as a reference. Cover Title page An Introduction to q-analysis Chapter 1. Inversions 1.1. Stern’s problem Exercises 1.2. The q-factorial Exercises 1.3. q-binomial coefficients Exercises 1.4. Some identities for q-binomial coefficients Exercises 1.5. Another property of q-binomial coefficients Exercises 1.6. q-multinomial coefficients Exercises 1.7. The Z-identity Exercises 1.8. Bibliographical Notes Chapter 2. q-binomial Theorems 2.1. A noncommutative q-binomial Theorem Exercises 2.2. Potter’s proof Exercises 2.3. Rothe’s q-binomial theorem Exercises 2.4. The q-derivative Exercises 2.5. Two q-binomial theorems of Gauss Exercises 2.6. Jacobi’s q-binomial theorem Exercises 2.7. MacMahon’s q-binomial theorem Exercises 2.8. A partial fraction decomposition Exercises 2.9. A curious q-identity of Euler, and some extensions Exercises 2.10. The Chen–Chu–Gu identity Exercises 2.11. Bibliographical Notes Chapter 3. Partitions I: Elementary Theory 3.1. Partitions with distinct parts Exercises 3.2. Partitions with repeated parts Exercises 3.3. Ferrers diagrams Exercises 3.4. q-binomial coefficients and partitions Exercises 3.5. An identity of Euler, and its “finite" form Exercises 3.6. Another identity of Euler, and its finite form Exercises 3.7. The Cauchy/Crelle q-binomial series Exercises 3.8. q-exponential functions Exercises 3.9. Bibliographical Notes Chapter 4. Partitions II: Geometric Theory 4.1. Euler’s pentagonal number theorem Exercises 4.2. Durfee squares Exercises 4.3. Euler’s pentagonal number theorem: Franklin’s proof Exercises 4.4. Divisor sums Exercises 4.5. Sylvester’s fishhook bijection Exercises 4.6. Bibliographical Notes Chapter 5. More q-identities: Jacobi, Gauss, and Heine 5.1. Jacobi’s triple product Exercises 5.2. Other proofs and related results Exercises 5.3. The quintuple product identity Exercises 5.4. Lebesgue’s identity Exercises 5.5. Basic hypergeometric series Exercises 5.6. More 2φ1 identities Exercises 5.7. The q-Pfaff–Saalschütz identity Exercises 5.8. Bibliographical Notes Chapter 6. Ramanujan’s 1ψ1 Summation Formula 6.1. Ramanujan’s formula Exercises 6.2. Four proofs Exercises 6.3. From the q-Pfaff–Saalschütz sum to Ramanujan’s 1ψ1 summation Exercises 6.4. Another identity of Cauchy, and its finite form Exercises 6.5. Cauchy’s “mistaken identity” Exercises 6.6. Ramanujan’s formula again Exercises 6.7. Bibliographical Notes Chapter 7. Sums of Squares 7.1. Cauchy’s formula Exercises 7.2. Sums of two squares Exercises 7.3. Sums of four squares Exercises 7.4. Bibliographical Notes Chapter 8. Ramanujan’s Congruences 8.1. Ramanujan’s congruences Exercises 8.2. Ramanujan’s “most beautiful" identity Exercises 8.3. Ramanujan’s congruences again 8.4. Bibliographical Notes Chapter 9. Some Combinatorial Results 9.1. Revisiting the q-factorial Exercises 9.2. Revisiting the q-binomial coefficients Exercises 9.3. Foata’s bijection for q-multinomial coefficients Exercises 9.4. MacMahon’s proof Exercises 9.5. q-derangement numbers Exercises 9.6. q-Eulerian numbers and polynomials Exercises 9.7. q-trigonometric functions Exercises 9.8. Combinatorics of q-tangents and secants 9.9. Bibliographical Notes Chapter 10. The Rogers–Ramanujan Identities I: Schur 10.1. Schur’s extension of Franklin’s argument Exercises 10.2. The Bressoud–Chapman proof Exercises 10.3. The AKP and GIS identities 10.4. Schur’s second partition theorem Exercises 10.5. Bibliographical Notes Chapter 11. The Rogers–Ramanujan Identities II: Rogers 11.1. Ramanujan’s proof Exercises 11.2. The Rogers–Ramanujan identities and partitions Exercises 11.3. Rogers’s second proof Exercises 11.4. More identities of Rogers Exercises 11.5. Rogers’s identities and partitions 11.6. The Göllnitz–Gordon identities Exercises 11.7. The Göllnitz–Gordon identities and partitions Exercises 11.8. Bibliographical Notes Chapter 12. The Rogers–Selberg Function 12.1. The Rogers–Selberg function Exercises 12.2. Some applications Exercises 12.3. The Selberg coefficients Exercises 12.4. The case k=3 12.5. Explicit formulas for the Q functions Exercises 12.6. Explicit formulas for S_{3,i}(x) Exercises 12.7. The payoff for k=3 Exercises 12.8. Gordon’s theorem 12.9. Bibliographical Notes Chapter 13. Bailey’s 6ψ6 Sum 13.1. Bailey’s formula Exercises 13.2. Another proof of Ramanujan’s “most beautiful" identity 13.3. Sums of eight squares and of eight triangular numbers Exercises 13.4. Bailey’s 6ψ6 summation formula Exercises 13.5. Askey’s proof: Phase 1 Exercises 13.6. Askey’s proof: Phase 2 Exercises 13.7. Askey’s proof: Phase 3 Exercises 13.8. An integral Exercises 13.9. Bailey’s lemma 13.10. Watson’s transformation Exercises 13.11. Bibliographical Notes Appendix A. A Brief Guide to Notation Appendix B. Infinite Products Exercises Appendix C. Tannery’s Theorem Bibliography 1-17 18-42 43-68 69-90 91-111 112-135 136-159 160-183 184-207 208-227 228-247 248-253 Index of Names Index of Topics Back Cover