An Invitation to Q-series: From Jacobi's Triple Product Identity to Ramanujan's "most Beautiful Identity"
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.
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Chapter 1 Introduction
Jacobis Triple Product Identity
The RogersRamanujan Identities
The RogersRamanujan Continued Fraction
From the Most Beautiful Identity to Ramanujans congruences
Appendix A Proofs of η 1τ iτ ητ
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An Invitation to Q-series: From Jacobi's Triple Product Identity to ...
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acobi’s triple product Andrews and Berndt Bailey’s lemma beautiful identity bijection Bressoud coeﬁicient column congruences crank cubic continued fraction cusp cusp form define Deﬁnition denote desired result Durfee square Dyson Engel expansion equivalent evaluate example Exercise 12.4 Question Exercise 4.1 Exercise 9.5 Question exposed cell factor fermions ﬁrst following identity functional equation Garvan Gaussian polynomials Golden Ratio H. H. Chan Hence Hirschhorn intentionally left blank Kim and Stanton last equation left blank Chapter left-hand side Let us deﬁne Lie algebra mock theta functions modular forms non-zero notation number of partitions partition function positive integer Prodinger 2000 proof of Eq proof of Theorem prove Eq prove the following prove Theorem q-binomial theorem q-series Ramanujan’s recall Remark right-hand side rim hook Rogers-Ramanujan identities second proof side of Eq Step t-cores triple product identity Young diagram Zwegers