Let (with ). As usual the Dedekind eta function is defined as
Then See [12, p. 121,] for a proof. Now for p>1 (a positive integer) we define where and . It should be noted that the functions and occured naturally in the Borwein-Borwein cubic iteration [6], [7]. Define where Here . From (3.2) we have and which follows from (3.3). The definition of was chosen so that it had a form analogous to that of (2.4) and that it satisfied a transformation like (3.9) below. Using (3.3) and (3.8) it is not hard to show that Substituting r=1 gives Since as we see thatlet be fixed. We have where and Further where
When N=p, the function has a nice form
so that The proof of (3.17) follows easily from (3.2), (3.4) and (3.15). From (3.8) and (3.14) we have By using (3.18) and (3.19) we find that (3.9) and (3.18) give rise to two equations involving and . These equations may be solved easily to yield