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