2 BERNDT, CHOI, CHOI, HAHN, YEAP, YEE, YESILYURT, YI
my belief is well-founded, the undivided credit for the discovery of these formulae
is due to Ramanujan." This last statement appears to be so obvious, especially
since the manuscript was evidently in Watson's possession, that one wonders why
he wrote it.
Ramanujan's forty identities for G(q) and H(q) (which do not include (1.2))
were first brought in their entirety before the mathematical public by B. J. Birch
[11], who in 1975 found Watson's handwritten copy of Ramanujan's list of forty
identities in the Oxford University Library. Ramanujan's original manuscript was
in Watson's possession for many years and now has evidently been lost. Watson's
handwritten list was later published along with Ramanujan's lost notebook [29,
pp. 236-237] in 1988. Certain pairs of the identities are linked, and so it is natural
to place them, in fact, in 35 (not 40) separate entries.
D. Bressoud [14], in his Ph.D. thesis, proved fifteen from the list of forty. His
published paper [15] contains proofs of some, but not all, of the general identities
from [14] which he developed in order to prove Ramanujan's identities. All the
proofs of Rogers, Watson, and Bressoud employ classical means, although it would
seem that in most cases the proofs are not like those found by Ramanujan.
After the work of Rogers, Watson, and Bressoud, nine remained to be proved.
A. J. F. Biagioli [10] used modular forms to prove eight of them. At this moment
then, only one of the forty identities has not been proved by any means, but it is
clear that modular forms can be used to establish this last identity. About such
proofs, Birch [11] opines, "A dull proof would have little value - in fact, all the
functions involved in the identities are essentially theta functions, so modular forms
of known level with poles of bounded order at known places, so the identities may
presumably be verified by just checking that the first hundred or so powers of x
are correct." It should be remarked that Biagioli's [10] proofs are more elegant
than one might discern from Birch's remarks, for Biagioli used Fricke involutions
and other properties of modular forms to drastically reduce the number of terms
envisioned by Birch. In fact, in most cases, Biagioli required only a few terms.
In this paper, we offer proofs of 35 of the 40 identities. Some of the proofs that
we present were found by either Rogers, Watson, or Bressoud. However, most of
the proofs presented in this paper are new. Frequently, we provide two or three
proofs of an identity. Our goal has been to find proofs for all forty identities
that Ramanujan might have given himself. Indeed, in several of our proofs, we
utilize modular equations found by Ramanujan and recorded in his notebooks [28].
Although all the proofs offered here are in the spirit of Ramanujan's mathematics,
it is to be admitted that for some proofs, knowing the identity beforehand was a
distinct advantage to us in finding a proof. It is unfortunate that we have failed to
find proofs of five of the identities, four of which were proved by Biagioli [10] using
the theory of modular forms and one of which has not been proved at all. However,
for each of these five identities, we offer heuristic arguments showing that both sides
of the identity have the same asymptotic expansions as q 1 —. It is very likely
that Ramanujan discovered many of his identities for G(q) and H(q) by examining
asymptotic expansions. Ramanujan was an expert on asymptotic expansions, and
in his last letter to G. H. Hardy written on January 12, 1920, Ramanujan discussed
the asymptotic expansions of his new mock theta functions and compared them to
the asymptotic expansion of G(q) with which he opened his letter [8, p. 220]. In
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