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Introduction to Q-analysis [Minkštas viršelis]

  • Formatas: Paperback / softback, 519 pages, weight: 945 g
  • Išleidimo metai: 30-Jan-2021
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470456230
  • ISBN-13: 9781470456238
Kitos knygos pagal šią temą:
Introduction to Q-analysisDidesnis paveikslėlis
  • Formatas: Paperback / softback, 519 pages, weight: 945 g
  • Išleidimo metai: 30-Jan-2021
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470456230
  • ISBN-13: 9781470456238
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781470456238.html
Raktažodžiai:
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.
An Introduction to q-analysis xi
Chapter 1 Inversions
1(38)
1.1 Stem's problem
1(6)
Exercises
5(2)
1.2 The g-factorial
7(7)
Exercises
11(3)
1.3 q-binomial coefficients
14(6)
Exercises
19(1)
1.4 Some identities for q-binomial coefficients
20(5)
Exercises
23(2)
1.5 Another property of g-binomial coefficients
25(4)
Exercises
27(2)
1.6 q-multinomial coefficients
29(4)
Exercises
31(2)
1.7 The Z-identity
33(4)
Exercises
36(1)
1.8 Bibliographical Notes
37(2)
Chapter 2 Gr-binomial Theorems
39(54)
2.1 A noncommutative q-binomial Theorem
39(6)
Exercises
43(2)
2.2 Potter's proof
45(4)
Exercises
47(2)
2.3 Rothe's g-binomial theorem
49(8)
Exercises
53(4)
2.4 The g-derivative
57(4)
Exercises
59(2)
2.5 Two q-binomial theorems of Gauss
61(10)
Exercises
66(5)
2.6 Jacobi's g-binomial theorem
71(3)
Exercises
72(2)
2.7 MacMahon's g-binomial theorem
74(5)
Exercises
77(2)
2.8 A partial fraction decomposition
79(3)
Exercises
82(1)
2.9 A curious (/-identity of Euler, and some extensions
82(6)
Exercises
86(2)
2.10 The Chen-Chu-Gu identity -
88(3)
Exercises
91(1)
2.11 Bibliographical Notes
91(2)
Chapter 3 Partitions I: Elementary Theory
93(56)
3.1 Partitions with distinct parts
93(5)
Exercises
95(3)
3.2 Partitions with repeated parts
98(8)
Exercises
103(3)
3.3 Ferrers diagrams
106(10)
Exercises
113(3)
3.4 q-binomial coefficients and partitions
116(4)
Exercises
119(1)
3.5 An identity of Euler, and its "finite" form
120(8)
Exercises
126(2)
3.6 Another identity of Euler, and its finite form
128(4)
Exercises
130(2)
3.7 The Cauchy/Crelle q-binomial series
132(9)
Exercises
137(4)
3.8 q-exponential functions
141(7)
Exercises
145(3)
3.9 Bibliographical Notes
148(1)
Chapter 4 Partitions II: Geometric Theory
149(42)
4.1 Euler's pentagonal number theorem
149(8)
Exercises
153(4)
4.2 Durfee squares
157(7)
Exercises
162(2)
4.3 Euler's pentagonal number theorem: Franklin's proof
164(3)
Exercises
167(1)
4.4 Divisor sums
167(13)
Exercises
173(7)
4.5 Sylvester's fishhook bijection
180(8)
Exercises
187(1)
4.6 Bibliographical Notes
188(3)
Chapter 5 More q-identities: Jacobi, Gauss, and Heine
191(56)
5.1 Jacobi's triple product
191(10)
Exercises
195(6)
5.2 Other proofs and related results
201(13)
Exercises
205(9)
5.3 The quintuple product identity
214(7)
Exercises
218(3)
5.4 Lebesgue's identity
221(6)
Exercises
223(4)
5.5 Basic hypergeometric series
227(6)
Exercises
230(3)
5.6 More q-identities
233(6)
Exercises
236(3)
5.7 The g-Pfaff Saalschutz identity
239(4)
Exercises
241(2)
5.8 Bibliographical Notes
243(4)
Chapter 6 Ramanujan's iipi Summation Formula
247(24)
6.1 Ramanujan's formula
247(3)
Exercises
249(1)
6.2 Four proofs
250(6)
Exercises
253(3)
6.3 From the q-Pfaff-Saalschutz sum to Ramanujan's 1ψ1 summation
256(3)
Exercises
259(1)
6.4 Another identity of Cauchy, and its finite form
259(4)
Exercises
260(3)
6.5 Cauchy's "mistaken identity"
263(3)
Exercises
265(1)
6.6 Ramanujan's formula again
266(2)
Exercises
268(1)
6.7 Bibliographical Notes
268(3)
Chapter 7 Sums of Squares
271(18)
7.1 Cauchy's formula
271(5)
Exercises
272(4)
7.2 Sums of two squares
276(5)
Exercises
278(3)
7.3 Sums of four squares
281(7)
Exercises
286(2)
7.4 Bibliographical Notes
288(1)
Chapter 8 Ramanujan's Congruences
289(16)
8.1 Ramanujan's congruences
289(3)
Exercises
291(1)
8.2 Ramanujan's "most beautiful" identity
292(8)
Exercises
298(2)
8.3 Ramanujan's congruences again
300(3)
8.4 Bibliographical Notes
303(2)
Chapter 9 Some Combinatorial Results
305(46)
9.1 Revisiting the q-factorial
305(6)
Exercises
309(2)
9.2 Revisiting the q-binomial coefficients
311(5)
Exercises
314(2)
9.3 Foata's bijection for q-multinomial coefficients
316(3)
Exercises
319(1)
9.4 MacMahon's proof
319(4)
Exercises
321(2)
9.5 g-derangement numbers
323(8)
Exercises
329(2)
9.6 Q-Eulerian numbers and polynomials
331(7)
Exercises
338(1)
9.7 Q-trigonometric functions
338(5)
Exercises
342(1)
9.8 Combinatorics of q-tangents and secants
343(6)
9.9 Bibliographical Notes
349(2)
Chapter 10 The Rogers-Ramanujan Identities I: Schur
351(26)
10.1 Schur's extension of Franklin's argument
351(6)
Exercises
356(1)
10.2 The Bressoud-Chapman proof
357(6)
Exercises
361(2)
10.3 The AKP and GIS identities
363(2)
10.4 Schur's second partition theorem
365(10)
Exercises
370(5)
10.5 Bibliographical Notes
375(2)
Chapter 11 The Rogers-Ramanujan Identities II: Rogers
377(40)
11.1 Ramanujan's proof
377(6)
Exercises
381(2)
11.2 The Rogers-Ramanujan identities and partitions
383(5)
Exercises
388(1)
11.3 Rogers's second proof
388(6)
Exercises
391(3)
11.4 More identities of Rogers
394(5)
Exercises
399(1)
11.5 Rogers's identities and partitions
399(4)
11.6 The Gollnitz-Gordon identities
403(9)
Exercises
407(5)
11.7 The Gollnitz-Gordon identities and partitions
412(4)
Exercises
414(2)
11.8 Bibliographical Notes
416(1)
Chapter 12 The Rogers-Selberg Function
417(20)
12.1 The Rogers-Selberg function
417(3)
Exercises
419(1)
12.2 Some applications
420(3)
Exercises
423(1)
12.3 The Selberg coefficients
423(4)
Exercises
427(1)
12.4 The case k = 3
427(2)
12.5 Explicit formulas for the Q functions
429(1)
Exercises
430(1)
12.6 Explicit formulas for S3,i(χ)
430(2)
Exercises
431(1)
12.7 The payoff for k = 3
432(2)
Exercises
434(1)
12.8 Gordon's theorem
434(2)
12.9 Bibliographical Notes
436(1)
Chapter 13 Bailey's 6&spi;6 Sum
437(64)
13.1 Bailey's formula
437(5)
Exercises
439(3)
13.2 Another proof of Ramanujan's "most beautiful" identity
442(2)
13.3 Sums of eight squares and of eight triangular numbers
444(5)
Exercises
447(2)
13.4 Bailey's 6ψ6 summation formula
449(5)
Exercises
450(4)
13.5 Askey's proof: Phase 1
454(3)
Exercises
457(1)
13.6 Askey's proof: Phase 2
457(3)
Exercises
460(1)
13.7 Askey's proof: Phase 3
460(5)
Exercises
461(4)
13.8 An integral
465(6)
Exercises
470(1)
13.9 Bailey's lemma
471(4)
13.10 Watson's transformation
475(6)
Exercises
479(2)
13.11 Bibliographical Notes
481(2)
Appendix A A Brief Guide to Notation
483(4)
Appendix B Infinite Products
487(8)
Exercises
491(4)
Appendix C Tannery's Theorem
495(6)
Bibliography 501(12)
Index of Names 513(4)
Index of Topics 517
Warren P. Johnson, Connecticut College, New London, CT, USA