that is a nice eta-product:This identity can be proved easily from results in Ramanujan's notebook. From (3.15), (3.16) we have by [1, Entry 1(iv), Chapter 20, p. 345,]. In the last step we have used the classical cubic modular equation for the eta-function: See [7, Corollary 2.5,]. Using (6.5), (6.6) may be written terms of either or (defined in (6.2), (6.3)), which were involved in the cubic iteration. We have
This time we let where , , are defined in (6.1)--(6.3). We define the following functions: Using the results of Section 6 it can be shown that where . We can also utilize the initial values for the cubic and obtain a nonic iteration.
Define sequences , , and by where and where Then converges nonically to .