Description
This unique book explores the world of q, known technically as basic hypergeometric series, and represents the authorâs personal and life-long studyâinspired by Ramanujanâof 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 Lagrangeâs four-squares theorem and Gaussâs two-squares theorem. Attention then turns to partitions and Ramanujanâs partition congruences. Several proofs of these are given throughout the 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 RogersâRamanujan identities and the RogersâRamanujan continued fraction, the famous âforty identitiesâ 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 âmysteriousâ partition theorem of H. Farkas and prove a conjecture of R.Wm. Gosper âwhich even ErdÅs couldnât do.â The book concludes with a look at Ramanujanâs remarkable tau function.
Table of Contents
Foreword.- Preface.- 1. Introduction.- 2. Jacobi's two-squares and four-squares theorems.- 3. Ramanujan's partition congruences.- 4. Ramanujan's partition congruencesâ 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) â¡ 0 (mod 11), and more.- 14. Partitions where even parts come in two colours.- 15. The RogersâRamanujan identities and the RogersâRamanujan continued fraction.- 16. The series expansion of the RogersâRamanujan continued fraction.- 17. The 2- and 4-dissections of Ramanujanâs continued fraction and its reciprocal.- 18. The series expansion of the Ramanujan-Gollnitz-Gordon continued fraction and its reciprocal.- 19. Jacobiâs âaequatio identica satis abstrusaâ.- 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 JordanâKronecker identity.- 29. Melhamâs 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. Somemodular 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.
