12 BERNDT, CHOI, CHOI, HAHN, YEAP, YEE, YESILYURT, YI
or when one or both of qr and qs are replaced by qr and qs, respectively, in
either (3.44) or (3.45) above.
Ramanujan's identities are remarkable for several reasons. The Rogers-
Ramanujan functions are associated with modular equations of degree 5 and q-
products with base
q5.
However, the "5" is missing on all the right sides of the
identities, except for Entries 3.3 and 3.21. One would expect to see in such iden-
tities theta functions with arguments
q5n,
for certain positive integers n, but such
functions do not appear! We have one explanation for this phenomenon. In our
heuristic "proofs" of five of the forty identities, we use the transformation formulas
(6.1.1) and (6.1.2) in Lemma 6.1 for, respectively, G(q) and H(q). Appearing in
these transformation formulas are l/cos(27r/5) and l/cos(47r/5) as multiplicative
factors in (6.1.1) and (6.1.2), respectively. Also note the appearances of fifth roots
of unity in the infinite products on both sides of (6.1.1) and (6.1.2). In Lemma 6.2,
we show that when these products are expanded into power series, only those pow-
ers with index congruent to either 0 or 1 modulo 5 survive. In fact, Lemma 6.2 is a
version of the 5-dissection of the generating function for cranks. Moreover, appear-
ing in the coefficients when the index n = 1 (mod 5) are cos(27r/5) and cos(47r/5),
respectively, for the products in (6.1.1) and (6.1.2). In particular, see (6.1.4) and
(6.1.5), respectively. Thus, when working with the power series of G(qa) and H(q@)
in the transformed variable q, considerable cancellation takes place leaving eventu-
ally only powers of 5. We then make a change of variable, replacing
qb
by, say, u,
and so the prominence of "5" disappears.
Next, observe that the right sides in almost all of the identities are expressed
entirely in terms of the modular function \ w r t h n o other theta function appearing.
We have no explanation for this phenomenon. It seems likely that the function \
played a more important role in Ramanujan's thinking than we are able to discern.
As we shall see in the proofs throughout the paper, some of the identities
are amenable to general techniques established either by Watson, Rogers, or the
authors. However, for those identities that are more difficult to prove (and there
are many), these ideas do not appear to be useable. It was unsettling for us to find
a proof of a certain identity with a great deal of effort and then discover that our
ideas were inapplicable to any of the remaining identities that we sought to prove.
In other words, each of the "hard" identities required an argument that seems to
apply to only that identity. Thus, the authors feel that if Ramanujan did indeed
have ironclad proofs for each of his identities, he had at least one key idea that
all researchers to date have missed. It seems likely that the function \ played an
important role in Ramanujan's primary idea(s). Each of the forty identities, in
principle, can be associated with modular equations of a certain degree. It happens
that for each such degree, Ramanujan recorded at least one modular equation of
that degree in his notebooks [28], [5]. We conjecture that Ramanujan utilized
modular equations to prove some of the forty identities in manners that we have
not been able to discern.
Before embarking on the proofs, we summarize here those proofs that we have
borrowed from others and those entries that we are unable to prove. The proofs of
Entries 3.18 and 3.28 that we give are due to Bressoud [14]. Our proof of Entry
3.34 is a modification of his proof [14]. Our proof of Entry 3.19 begins at the same
point as that of Bressoud but diverges thereafter. We give two proofs of Entry
3.12, one of which is due to Bressoud [14]. Watson's proofs of Entries 3.3, 3.21,
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