q-Series

In view of Chapter 16 (Ramanujan [22], Berndt [6]), wherein several theorems on q-series are offered, and of the “lost notebook” (Ramanujan [24]), which is almost completely devoted to q-series, it is surprising that the unorganized portion of the second notebook and the third notebook contain only a small amount of material on q-series. In fact, only five identities are to be found here (Entries 1–5 below). More interesting are the asymptotic formulas given in Entries 6–8. Thus, in this chapter, only eight results are proved. Additional q-analysis can be found in Chapter 32, entitled Continued Fractions (Berndt [9]), or in the memoir of Andrews, Berndt, Jacobsen, and Lamphere [1]. Further related material can be found in Chapter 25 on theta-functions.