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El. knyga: Power of q: A Personal Journey

  • Formatas: EPUB+DRM
  • Serija: Developments in Mathematics 49
  • Išleidimo metai: 08-Aug-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319577623
Kitos knygos pagal šią temą:
Power of q: A Personal JourneyDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Developments in Mathematics 49
  • Išleidimo metai: 08-Aug-2017
  • Leidėjas: Springer International Publishing AG
  • Kalba: eng
  • ISBN-13: 9783319577623
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This unique book explores the world of q , known technically as basic hypergeometric series, and represents the author"s personal and life-long study-inspired by Ramanujan-of aspects of this broad topic. While the level of mathematical sophistication is graduated, the book is designed to appeal to advanced undergraduates as well as researchers in the field. The principal aims are to demonstrate the power of the methods and the beauty of the results. The book contains novel proofs of many results in the theory of partitions and the theory of representations, as well as associated identities. Though not specifically designed as a textbook, parts of it may be presented in course work; it has many suitable exercises.After an introductory chapter, the power of q -series is demonstrated with proofs of Lagrange"s four-squares theorem and Gauss"s two-squares theorem. Attention then turns to partitions and Ramanujan"s partition congruences. Several proofs of these are given throughout th

e book. Many chapters are devoted to related and other associated topics. One highlight is a simple proof of an identity of Jacobi with application to string theory. On the way, we come across the Rogers-Ramanujan identities and the Rogers-Ramanujan continued fraction, the famous "forty identities" of Ramanujan, and the representation results of Jacobi, Dirichlet and Lorenz, not to mention many other interesting and beautiful results. We also meet a challenge of D.H. Lehmer to give a formula for the number of partitions of a number into four squares, prove a "mysterious" partition theorem of H. Farkas and prove a conjecture of R.Wm. Gosper "which even Erdös couldn"t do." The book concludes with a look at Ramanujan"s remarkable tau function.

Foreword.- Preface.- 1. Introduction.- 2. Jacobi"s two-squares and four-squares theorems.- 3. Ramanujan"s partition congruences.- 4. Ramanujan"s partition congruences- a uniform proof.- 5. Ramanujan"s "most beautiful identity".- 6. Ramanujan"s partition congruences for powers of 5.- 7. Ramanujan"s partition congruences for powers of 7.- 8. Ramanujan"s 5-dissection of Euler"s product.- 9. A "difficult and deep" identity of Ramanujan.- 10. The quintuple product identity.- 11. Winquist"s identity.- 12. The crank of a partition.- 13. Two more proofs of p(11n + 6) 0 (mod 11), and more.- 14. Partitions where even parts come in two colours.- 15. The Rogers-Ramanujan identities and the Rogers-Ramanujan continued fraction.- 16. The series expansion of the Rogers-Ramanujan continued fraction.- 17. The 2- and 4-dissections of Ramanujan"s continued fraction and its reciprocal.- 18. The series expansion of the Ramanujan-Gollnitz-Gordon continued fraction and its reciprocal.- 19. Jacobi"s " aeq

uatio identica satis abstrusa ".- 20. Two modular equations.- 21. A letter from Fitzroy House.- 22. The cubic functions of Borwein, Borwein and Garvan.- 23. Some classical results on representations.- 24. Further classical results on representations.- 25. Further results on representations.- 26. Even more representation results.- 27. Representation results and Lambert series.- 28. The Jordan-Kronecker identity.- 29. Melham"s identities.- 30. Partitions into four squares.- 31. Partitions into four distinct squares of equal parity.- 32. Partitions with odd parts distinct.- 33. Partitions with even parts distinct.- 34. Some identities involving phi( q ) and psi( q ).- 35. Some useful parametrisations.- 36. Overpartitions.- 37. Bipartitions with odd parts distinct.- 38. Overcubic partitions.- 39. Generalised Frobenius partitions.- 40. Some modular equations of Ramanujan.- 41. Identities involving k = qR(q)R(q2)2 .- 42. Identities involving v=q1/2(q,q7;q8)infinity/(q3,q5;q8)infinity .-

43. Ramanujan"s tau function.- Appendix.- Index.

Recenzijos

This book provides an introduction to q-series that would be accessible to calculus students, its main purpose is to offer beautiful theorems to the reader along with, in many instances, equally beautiful proofs that cannot be found elsewhere, except possibly in the authors own papers. those who already love q-series will find much to admire and enjoy in Hirschhorns book The Power of q. Those desiring an introduction to the subject can also enjoy it. (Bruce Berndt, The American Mathematical Monthly, Vol. 126 (2), April, 2019)

1 Introduction
1(18)
1.1 Introduction
1(1)
1.2 An Identity of Euler
2(1)
1.3 Our First Proof of Jacobi's Triple Product Identity
3(2)
1.4 Our Second Proof of Jacobi's Triple Product Identity
5(3)
1.5 Some Important Special Cases of Jacobi's Triple Product Identity
8(2)
1.6 Euler's Pentagonal Numbers Theorem
10(1)
1.7 Jacobi's Formula for the Cube of Euler's Product
11(1)
1.8 Polynomial Versions of Earlier Results
12(2)
1.9 m-Dissection
14(1)
1.10 Linear Transformations
15(4)
2 Jacobi's Two-Squares and Four-Squares Theorems
19(8)
2.1 Introduction
19(1)
2.2 Our First Proof of Jacobi's Two-Squares Theorem
20(2)
2.3 Our Second Proof of Jacobi's Two-Squares Theorem
22(1)
2.4 A Proof of Jacobi's Four-Squares Theorem
23(4)
3 Ramanujan's Partition Congruences
27(16)
3.1 Introduction
27(3)
3.2 p(5n + 4) ≡ 0(mod 5)
30(2)
3.3 p(5n + 4) ≡ 0(mod 5) Again
32(1)
3.4 p(7n + 5) ≡ 0(mod 7)
32(1)
3.5 p(11n + 6) ≡ 0 (mod 11)
33(2)
3.6 The Atkin--Swinnerton-Dyer Congruences for the Modulus 5
35(2)
3.7 The Atkin--Swinnerton-Dyer Congruences for the Modulus 7
37(2)
3.8 The Atkin--Swinnerton-Dyer Congruences for the Modulus 11
39(4)
4 Ramanujan's Partition Congruences---A Uniform Proof
43(12)
4.1 Introduction
43(1)
4.2 p(5n + 4) ≡ 0(mod 5)
43(5)
4.3 p(7n + 5) ≡ 0(mod 7)
48(3)
4.4 p(11n + 6) ≡ 0 (mod 11)
51(4)
5 Ramanujan's Most Beautiful Identity
55(4)
5.1 Introduction
55(1)
5.2 The 5-Dissection of the Partition Generating Function
55(2)
5.3 Two Surprising Results
57(1)
5.4 Ramanujan's Most Beautiful Identity
58(1)
6 Ramanujan's Partition Congruences for Powers of 5
59(12)
6.1 Introduction
59(1)
6.2 The Modular Equation of Degree 5
59(2)
6.3 Ramanujan's Most Beautiful Identity Again
61(1)
6.4 The Generating Function Σn≥0 p(5αn + δx)qn
62(4)
6.5 Ramanujan's Partition Congruences for Powers of 5
66(1)
6.6 A Conjecture of Ramanujan
67(1)
6.7 Some Comments
68(3)
7 Ramanujan's Partition Congruences for Powers of 7
71(14)
7.1 Introduction
71(1)
7.2 The 7-Dissection of Euler's Product
71(2)
7.3 The 7-Analogue of Ramanujan's Most Beautiful Identity
73(3)
7.4 The Modular Equation of Degree 7
76(2)
7.5 Ramanujan's Partition Congruences for Powers of 7
78(4)
7.6 Two More Formulas
82(3)
8 Ramanujan's 5-Dissection of Euler's Product
85(8)
8.1 Introduction
85(1)
8.2 Our First Proof of Ramanujan's 5-Dissection of Euler's Product
86(1)
8.3 Our Second Proof of Ramanujan's 5-Dissection of Euler's Product
87(1)
8.4 Review of Earlier Results
88(1)
8.5 A Factorisation of Identity (8.4.3)
89(2)
8.6 Ramanujan's Versions of (8.3.1) and (8.3.2), (8.5.4) and (8.5.5)
91(2)
9 A "Difficult and Deep" Identity of Ramanujan
93(6)
9.1 Introduction
93(1)
9.2 Proof Inspired by Chadwick Gugg
93(2)
9.3 Proof Inspired by G.N. Watson
95(4)
10 The Quintuple Product Identity
99(10)
10.1 Introduction
99(1)
10.2 Proof of the Quintuple Product Identity
100(1)
10.3 A Generalisation of the Quintuple Product Identity
101(1)
10.4 Ramanujan's 5-Dissection of Euler's Product Again
102(1)
10.5 The 7-Dissection of Euler's Product
103(1)
10.6 The 11-Dissection of Euler's Product
103(1)
10.7 Two Identities of Ramanujan
104(1)
10.8 Continuation
105(3)
10.9 Some Interesting Results
108(1)
11 Winquist's Identity
109(4)
11.1 Introduction
109(1)
11.2 A Proof of Winquist's Identity
110(3)
12 The Crank of a Partition
113(10)
12.1 Introduction
113(1)
12.2 Introducing the Crank
114(2)
12.3 Proofs of (12.2.9)---(12.2.11)
116(4)
12.4 The Definition of the Crank
120(3)
13 Two More Proofs of p(11n + 6) ≡ 0 (mod 11), and More
123(8)
13.1 Introduction
123(1)
13.2 The Expansion of E(q)10
124(1)
13.3 p(11n + 6) ≡ 0(mod 11) Again
125(3)
13.4 The Atkin--Swinnerton-Dyer Congruences for Modulus 11
128(3)
14 Partitions Where Even Parts Come in Two Colours
131(8)
14.1 Introduction
131(1)
14.2 The Generating Function for the p*(n)
131(1)
14.3 The 3-Dissection of F(q)
132(1)
14.4 Hei-Chi Chan's Most Beautiful Identity?
132(1)
14.5 The Modular Equation of Degree 3
133(2)
14.6 The Generating Function Σn≥0 p*(3αn + δα)qn
135(1)
14.7 Hei-Chi Chan's Partition Congruences
136(1)
14.8 Some New Results
137(2)
15 The Rogers--Ramanujan Identities and the Rogers--Ramanujan Continued Fraction
139(10)
15.1 Introduction
139(2)
15.2 Ramanujan's Proof of the Rogers--Ramanujan Identities
141(2)
15.3 Andrews' Proof of the Rogers--Ramanujan Identities
143(1)
15.4 The Rogers--Ramanujan Continued Fraction (Ramanujan's Original Derivation)
143(1)
15.5 The Rogers--Ramanujan Continued Fraction (Second Derivation)
144(2)
15.6 A Finite Version of the Rogers--Ramanujan Continued Fraction
146(1)
15.7 The "Difficult and Deep" Identity of
Chapter 9
147(2)
16 The Series Expansion of the Rogers--Ramanujan Continued Fraction and Its Reciprocal
149(8)
16.1 Introduction
149(1)
16.2 Expanding the Rogers--Ramanujan Continued Fraction and Its Reciprocal
150(1)
16.3 Proofs of Szekeres's Observations
150(7)
17 The 2- and 4-Dissections of the Rogers--Ramanujan Continued Fraction and Its Reciprocal
157(6)
17.1 Introduction
157(1)
17.2 The 2-Dissections
158(1)
17.3 The 4-Dissections
159(1)
17.4 Forty Identities of Ramanujan
160(3)
18 The Series Expansion of the Ramanujan--Gollnitz--Gordon Continued Fraction and Its Reciprocal
163(6)
18.1 Introduction
163(1)
18.2 The 2-, 4- and 8-dissections
164(5)
19 Jacobi's "aequatio identica satis abstrusa"
169(6)
19.1 Introduction
169(1)
19.2 Our First Proof of Jacobi's identica abstrusa
169(2)
19.3 Our Second Proof of identica abstrusa
171(1)
19.4 Some Remarkable Formulations
171(3)
19.5 A Simple Generalisation of Jacobi's identica abstrusa
174(1)
20 Two Modular Equations
175(4)
20.1 Introduction
175(1)
20.2 Proof of the First Modular Equation
175(1)
20.3 Proof of the Second Modular Equation
176(1)
20.4 Combinatorial Interpretations
177(2)
21 A Letter from Fitzroy House
179(6)
21.1 Introduction
179(1)
21.2 Proof of the Lorenz--Ramanujan Identity (21.1.1)
179(4)
21.3 Further Identities Involving Σ∞m,n=-∞ qm2 + mn + n2
183(32)
22 The Cubic Theta-Function Analogues of Borwein, Borwein and Garvan
185(20)
22.1 Introduction
185(3)
22.2 The 3-Dissections of a{q) and b(q)
188(1)
22.3 Expression of b(q) and c(q) as Products
189(1)
22.4 Expressions for b(q) and c(q) Analogous to (21.1.1)
190(1)
22.5 A Fine Recurrence for p(n)
191(1)
22.6 The 2-Dissections of a(q), b(q) and q-1/3c(q)
192(3)
22.7 Two Further Relations Between b(q) and c(q)
195(1)
22.8 Two Relations Between a(q), b(q) and c(q)
196(3)
22.9 Partitions Where Even Parts Come in Two Colours
199(1)
22.10 Two Identities
200(1)
22.11 Some More Interesting Results
201(4)
23 Some Classical Results on Representations
205(6)
23.1 Introduction
205(1)
23.2 Proof of the Two Triangles Result (23.1.2)
206(1)
23.3 Square + 3 × Square
206(2)
23.4 Triangle + 3 × Triangle
208(1)
23.5 Our Second Proof of the Modular Equation (20.1.1)
209(2)
24 Further Classical Results on Representations
211(6)
24.1 Introduction
211(1)
24.2 Dirichlet's Result for Square + 2 × Square
212(1)
24.3 Our Third Proof of Jacobi's Two-Squares Theorem
212(1)
24.4 Our Second Proof of Jacobi's Four-Squares Theorem
213(1)
24.5 Our Second Proof of the Square + 3 × Square Result
214(1)
24.6 Proof of Lorenz's Partial Fractions Result
214(3)
25 Further Results on Representations
217(8)
25.1 Introduction
217(2)
25.2 Preliminary Result
219(1)
25.3 Our Third Proof of the Modular Equation (20.1.1)
220(1)
25.4 Proofs of the Representation Results (25.1.12)---(25.1.28)
220(5)
26 Even More Representation Results
225(4)
26.1 Introduction
225(2)
26.2 Proofs
227(2)
27 Representation Results and Lambert Series
229(6)
27.1 Introduction
229(2)
27.2 Statements of Lambert Series and Bilateral Lambert Series Results
231(4)
28 The Jordan--Kronecker Identity
235(12)
28.1 Introduction
235(1)
28.2 The Jordan--Kronecker Identity
235(2)
28.3 The Two Triangles Result
237(1)
28.4 Triangle + 2 × Triangle
238(1)
28.5 Triangle + 3 × Triangle
238(1)
28.6 Triangle + 4 × Triangle
238(1)
28.7 Triangle + Square
239(1)
28.8 Triangle + 2 × Square
240(1)
28.9 2 × Triangle + 3 × Square
241(2)
28.10 4 × Triangle + Square
243(1)
28.11 6 × Triangle + Square
243(2)
28.12 Our Fourth Proof of the Modular Equation (20.1.1)
245(1)
28.13 Proof of (22.11.5)
245(2)
29 Melham's Identities
247(10)
29.1 Introduction
247(1)
29.2 Triangle + 5 × Triangle
248(3)
29.3 Triangle + 6 × Triangle
251(2)
29.4 Pentagon + 5 × Pentagon
253(4)
30 Partitions into Four Squares
257(32)
30.1 Introduction
257(1)
30.2 The Generating Functions
257(4)
30.3 Some Relations
261(3)
30.4 An Important Corollary
264(1)
30.5 Some Congruences
265(2)
30.6 The Generating Function for a(72n + 69)
267(2)
30.7 A Remarkable Identity
269(2)
30.8 The Generating Function for b(72n + 69)
271(3)
30.9 The Generating Function for c(72n + 69)
274(4)
30.10 Some Important and Useful 2-Dissections
278(3)
30.11 The Final Steps in the Proofs of (30.5.5)---(30.5.8)
281(3)
30.12 Our Fifth Proof of the Modular Equation (20.1.1)
284(1)
30.13 A Formula for p4□(n)
285(4)
31 Partitions into Four Distinct Squares of Equal Parity
289(8)
31.1 Introduction
289(1)
31.2 The Generating Function for pd4o (8n + 4)
290(3)
31.3 The Generating Function for pd4e (8n + 4) +pd+4e (8n + 4)
293(1)
31.4 Completing the Proof of (31.1.2)
294(1)
31.5 Some More Interesting Facts
294(3)
32 Partitions with Odd Parts Distinct
297(6)
32.1 Introduction
297(1)
32.2 The Generating Function for pod(n)
297(1)
32.3 Some Congruences
298(2)
32.4 Proof of the Congruence (32.1.1)
300(3)
33 Partitions with Even Parts Distinct
303(8)
33.1 Introduction
303(1)
33.2 The Generating Function for ped(n) and Other Preliminaries
303(2)
33.3 The Derivation of (33.1.3)
305(1)
33.4 Some Congruences
306(2)
33.5 Proofs of the Congruences (33.1.1) and (33.1.2)
308(3)
34 Some Identities Involving φ(q) and ψ(q)
311(24)
34.1 Introduction
311(3)
34.2 Proofs of (34.1.1) and (34.1.2)
314(2)
34.3 Proofs of (34.1.3) and (34.1.4)
316(3)
34.4 Proofs of (34.1.5) and (34.1.6)
319(2)
34.5 Proofs of (34.1.7) and (34.1.8)
321(4)
34.6 Proofs of (34.1.9) and (34.1.10)
325(2)
34.7 Proofs of (34.1.11) and (34.1.12)
327(2)
34.8 Proofs of (34.1.13)---(34.1.23)
329(2)
34.9 Proofs of (34.1.24) and (34.1.25)
331(1)
34.10 The 2n-dissection of Euler's Product
331(4)
35 Some Useful Parametrisations
335(4)
35.1 Introduction
335(1)
35.2 The Derivations
336(3)
36 Overpartitions
339(6)
36.1 Introduction
339(1)
36.2 Congruences for -p(n) Modulo Powers of 2
340(1)
36.3 Our First Proof that -p(40n + 35) = 0 (mod 40)
341(2)
36.4 Our Second Proof that -p(40n + 35) = 0 (mod 40)
343(2)
37 Bipartitions with Odd Parts Distinct
345(6)
37.1 Introduction
345(1)
37.2 The Mod 3 Results
346(2)
37.3 The Mod 5 Results
348(3)
38 Overcubic Partitions
351(6)
38.1 Introduction
351(1)
38.2 The Generating Function and First Steps
352(1)
38.3 Completion
353(4)
39 Generalised Frobenius Partitions
357(8)
39.1 Introduction
357(1)
39.2 Proofs of (39.1.1) and (39.1.2)
358(3)
39.3 Proof of (39.1.3)
361(1)
39.4 Proof of (39.1.4)
362(3)
40 Some Modular Equations of Ramanujan
365(8)
40.1 Introduction
365(1)
40.2 Proofs of (40.1.4)---(40.1.11)
366(2)
40.3 Proof of (40.1.12)
368(5)
41 Identities Involving k = r(q)r(q2)2
373(10)
41.1 Introduction
373(2)
41.2 Proofs of (41.1.2) and (41.1.3)
375(1)
41.3 Proof of (41.1.4)
376(1)
41.4 Proof of (41.1.5)
376(1)
41.5 Proof of (41.1.6)
376(1)
41.6 Proofs of (41.1.7) and (41.1.8)
377(1)
41.7 Proofs of (41.1.9) and (41.1.10)
377(3)
41.8 Proofs of (41.1.11) and (41.1.12)
380(3)
42 Identities Involving v = q1/2(q, q7; q8)∞/(q3, q5; q8)∞
383(10)
42.1 Introduction
383(2)
42.2 Proofs of (42.1.2), (42.1.9) and (42.1.10)
385(2)
42.3 Proofs of (42.1.3), (42.1.11) and (42.1.12)
387(2)
42.4 Proofs of (42.1.4), (42.1.5) and (42.1.6)
389(1)
42.5 Proofs of (42.1.13) and (42.1.14)
390(2)
42.6 Proofs of (42.1.15) and (42.1.16), (42.1.7) and (42.1.8)
392(1)
43 Ramanujan's Tau Function
393(8)
43.1 Introduction
393(1)
43.2 p = 2
393(3)
43.3 p = 3
396(2)
43.4 p = 5
398(1)
43.5 p = 7
399(2)
Appendix 401(4)
References 405(6)
Index 411
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