An Introduction to q-analysis

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Title page
An Introduction to ?-analysis
Chapter 1. Inversions
	1.1. Stern’s problem
	Exercises
	1.2. The ?-factorial
	Exercises
	1.3. ?-binomial coefficients
	Exercises
	1.4. Some identities for ?-binomial coefficients
	Exercises
	1.5. Another property of ?-binomial coefficients
	Exercises
	1.6. ?-multinomial coefficients
	Exercises
	1.7. The Z-identity
	Exercises
	1.8. Bibliographical Notes
Chapter 2. ?-binomial Theorems
	2.1. A noncommutative ?-binomial Theorem
	Exercises
	2.2. Potter’s proof
	Exercises
	2.3. Rothe’s ?-binomial theorem
	Exercises
	2.4. The ?-derivative
	Exercises
	2.5. Two ?-binomial theorems of Gauss
	Exercises
	2.6. Jacobi’s ?-binomial theorem
	Exercises
	2.7. MacMahon’s ?-binomial theorem
	Exercises
	2.8. A partial fraction decomposition
	Exercises
	2.9. A curious ?-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. ?-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 ?-binomial series
	Exercises
	3.8. ?-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 ?-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 ₂?₁ identities
	Exercises
	5.7. The ?-Pfaff–Saalschütz identity
	Exercises
	5.8. Bibliographical Notes
Chapter 6. Ramanujan’s ₁?₁ Summation Formula
	6.1. Ramanujan’s formula
	Exercises
	6.2. Four proofs
	Exercises
	6.3. From the ?-Pfaff–Saalschütz sum to Ramanujan’s ₁?₁ 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 ?-factorial
	Exercises
	9.2. Revisiting the ?-binomial coefficients
	Exercises
	9.3. Foata’s bijection for ?-multinomial coefficients
	Exercises
	9.4. MacMahon’s proof
	Exercises
	9.5. ?-derangement numbers
	Exercises
	9.6. ?-Eulerian numbers and polynomials
	Exercises
	9.7. ?-trigonometric functions
	Exercises
	9.8. Combinatorics of ?-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 ?=3
	12.5. Explicit formulas for the ? functions
	Exercises
	12.6. Explicit formulas for ?_{3,?}(?)
	Exercises
	12.7. The payoff for ?=3
	Exercises
	12.8. Gordon’s theorem
	12.9. Bibliographical Notes
Chapter 13. Bailey’s ₆?₆ 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 ₆?₆ 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
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