EAN: 9783319577616

THE POWER OF Q - MICHAEL D HIRSCHHORN GEBUNDEN

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 Erdös couldn't do. The book concludes with a
look at Ramanujan's remarkable tau function.

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