xviii B. C. BERND T at various arguments in terms of certain elliptic parameters. Our exposition is derived from Chapter 17 of Ramanujan's second note- book [193], [34, Chapter 17], where, through a series of preliminary lemmas, Ramanujan leads us to the aforementioned key theorem. Applications of the aforementioned representations for Eisenstein series and theta functions form the content of Chapter 6. First, we return to the topic of sums of squares and demonstrate how the for- mulas for Eisenstein series lead to short proofs of some of the results from Chapter 3. However, most of Chapter 6 is devoted to modu- lar equations, a topic to which Ramanujan made more contributions than any other mathematician. Chapters 19-21 in his second note- book are devoted to modular equations, and our short introduction to this topic is drawn from our previous account of Ramanujan's work in these chapters [34]. One of Ramanujan's favorite topics was the Rogers-Ramanujan continued fraction, the focus of Chapter 7. Because we wish to share so much about this continued fraction with readers and because the length of the chapter would be prohibitive if we proved all theorems offered in this chapter, we forego some of the proofs. However, we do prove two key theorems relating the continued fraction with its recip- rocal. These theorems are then used to give an alternative, cleaner proof of an identity of Ramanujan in Chapter 2, yielding immedi- ately the congruence p(5n + 4) = 0(mod5). The famous Rogers- Ramanujan functions are also discussed, and, in particular, we prove that the Rogers-Ramanujan continued fraction can be represented as a quotient of the two Rogers-Ramanujan functions. Our fervent wish is that our sampling of the many beautiful properties satisfied by this continued fraction will motivate readers to turn to original sources to learn more about it. Ubiquitous in this book are products of the form (1 - a)(l - aq)(l - aq2) (1 - aqn~l) =: (a q)n as well as their infinite versions (1 - a)(l - aq)(l - aq2) (1 - aqn) =: (a q)^ \q\ 1, which are called ^-products. Although we assume that readers of this book are familiar with infinite series, it may well be that some
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