Theta Function Identities

Theta function identities are ubiquitous in Ramanujan’s notebooks [243]; in particular, see Berndt’s books [54] and [57] for several hundred such identities. Theta function identities are also prominent in Ramanujan’s lost notebook [244]. Several of these identities are intimately connected with the Rogers– Ramanujan continued fraction, and so these were examined in Chapter 1 of [31]. However, other chapters in [31] contain theta function identities, with Chapters 4, 13, 15, and 17 being exceptionally fruitful sources. Readers having read the first seven chapters of the present volume have seen how theta functions make perhaps unexpected appearances in q-series identities. Readers who continue to read the remainder of this volume will observe that theta functions and their identities are inextricably intertwined with Eisenstein series.