An invitation to q-series : from Jacobi's triple product identity to Ramanujan's "most beautiful identity&quot

The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers-Ramanujan continued fraction -- a result that convinced G H Hardy that Ramanujan was a "mathematician of the highest class", and (2) what G. H. Hardy called Ramanujan's "Most Beautiful Identity". This book covers a range of related results, such as several proofs of the famous Rogers-Ramanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series. 1. Introduction -- pt. I. Jacobi's triple product identity. 2. First proof (via functional equation). 3. Second proof (via Gaussian polynomials and the q-binomial theorem). 4. Some applications. 5. The Boson-Fermion correspondence. 6. Macdonald's identities -- pt. II. The Rogers-Ramanujan identities. 7. First proof (via functional equation). 8. Second proof (involving Gaussian polynomials and difference equations). 9. Third proof (via Bailey's lemma). 10. Excursus : Mock theta functions -- pt. III. The Rogers-Ramanujan continued fraction. 11. A list of theorems to be proven. 12. The evaluation of the Rogers-Ramanujan continued fraction. 13. A "difficult and deep" identity. 14. A remarkable identity from the Lost Notebook and cranks. 15. A differential equation for the Rogers-Ramanujan continued fraction -- pt. IV. From the "most beautiful identity" to Ramanujan's congruences. 16. Proofs of the "most beautiful identity" 17. Ramanujan's congruences I : analytical methods. 18. Ramanujan's congruences II : an introduction to t-cores. 19. Ramanujan's congruences III : more congruences. 20. Excursus : modular forms and more congruences for the partition function The aim of this lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes (1) the evaluation of the RogersRamanujan continued fraction a result that convinced G H Hardy that Ramanujan was a mathematician of the highest class, and (2) what G H Hardy called Ramanujan's Most Beautiful Identity. This book covers a range of related results, such as several proofs of the famous RogersRamanujan identities and a detailed account of Ramanujan's congruences. It also covers a range of techniques in q-series. Provides a self-contained exposition of several formulas discovered by Srinivasa Ramanujan. This book covers: the evaluation of the Rogers-Ramanujan continued fraction - a result that convinced G H Hardy that Ramanujan was a mathematician of the 'highest class', and what G H Hardy called Ramanujan's 'Most Beautiful Identity'. Hei-chi Chan. Includes Bibliographical References (p. 213-223) And Index.