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Field name	Details
Dewey Class	512.7
Title	The Power of q ([EBook]) : A Personal Journey / / by Michael D. Hirschhorn.
Author	Hirschhorn, Michael D.
Other name(s)	SpringerLink (Online service)
Publication	Cham : : Springer International Publishing : : Imprint: Springer, , 2017.
Physical Details	XXIV, 415 p. : online resource.
Series	Developments in mathematics 1389-2177 ; ; 49
ISBN	9783319577623
Summary Note	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 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 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 Erdos couldn?t do.? The book concludes with a look at Ramanujan?s remarkable tau function.:
Contents note	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.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-3-319-57762-3
Links to Related Works	Subject References: Mathematics . Number Theory . Authors: Hirschhorn, Michael D. . Corporate Authors: SpringerLink (Online service) . Series: Developments in mathematics . Classification: 512.7 .

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The Power of q: A Personal Journey

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