The Power of Q: A Personal Journey

ISBN-13: 9783319577616 / Angielski / Twarda / 2017 / 415 str.

Michael D. Hirschhorn

Opis

This unique book explores the world of q, known technically as basic hypergeometricseries, and represents the author's personal and life-long study--inspired by Ramanujan--ofaspects of this broad topic. While the level of mathematical sophistication isgraduated, the book is designed to appeal to advanced undergraduates as well asresearchers in the field. The principal aims are to demonstrate the power of the methodsand the beauty of the results. The book contains novel proofs of many results in thetheory 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 coursework; it has many suitable exercises.After an introductory chapter, the power of q-series is demonstrated with proofs ofLagrange's four-squares theorem and Gauss's two-squares theorem. Attention thenturns to partitions and Ramanujan's partition congruences. Several proofs of these aregiven throughout the book. Many chapters are devoted to related and other associatedtopics. One highlight is a simple proof of an identity of Jacobi with application tostring theory. On the way, we come across the Rogers-Ramanujan identities and theRogers-Ramanujan continued fraction, the famous "forty identities" of Ramanujan, andthe representation results of Jacobi, Dirichlet and Lorenz, not to mention many otherinteresting and beautiful results. We also meet a challenge of D.H. Lehmer to give aformula 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 evenErdős couldn't do." The book concludes with a look at Ramanujan's remarkable taufunction.

Recenzje

"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 author's own papers. ... those who already love q-series will find much to admire and enjoy in Hirschhorn's 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)

Spis treści

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 “aequatio 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.