Abstract
Sir Arthur Conan Doyle's famous fictional detective Sherlock Holmes and his
sidekick Dr. Watson go camping and pitch their tent under the stars. During the
night, Holmes wakes his companion and says, "Watson, look up at the stars and tell
me what you deduce." Watson says, "I see millions of stars, and it is quite likely
that a few of them are planets just like Earth. Therefore there may also be life on
these planets." Holmes replies, "Watson, you idiot. Somebody stole our tent."
When seeking proofs of Ramanujan's identities for the Rogers-Ramanujan func-
tions, Watson, i.e., G. N. Watson, was not an "idiot." He, L. J. Rogers, and
D. M. Bressoud found proofs for several of the identities. A. J. F. Biagioli de-
vised proofs for most (but not all) of the remaining identities. Although some
of the proofs of Watson, Rogers, and Bressoud are likely in the spirit of those
found by Ramanujan, those of Biagioli are not. In particular, Biagioli used the
theory of modular forms. Haunted by the fact that little progress has been made
into Ramanujan's insights on these identities in the past 85 years, the present
authors sought "more natural" proofs. Thus, instead of a missing tent, we have
had missing proofs, i.e., Ramanujan's missing proofs of his forty identities for the
Rogers-Ramanujan functions. In this paper, for 35 of the 40 identities, we offer
proofs that are in the spirit of Ramanujan. Some of the proofs presented here are
due to Watson, Rogers, and Bressoud, but most are new. Moreover, for several
identities, we present two or three proofs. For the five identities that we are unable
to prove, we provide non-rigorous verifications based on an asymptotic analysis of
the associated Rogers-Ramanujan functions. We think that this method, which is
related to the 5-dissection of the generating function for cranks found in Ramanu-
jan's lost notebook, is what Ramanujan might have used to discover several of the
more difficult identities. Some of the new methods in this paper can be employed
to establish new identities for the Rogers-Ramanujan functions.
Received by the editor on January 28, 2005.
1991 Mathematics Subject Classification. 11P82, 11F27, 33D15.
Key words and phrases. Rogers-Ramanujan functions, Ramanujan's lost notebook, theta
functions, Rogers's method, modular equations, transformation formulas, asymptotic expansions
of Rogers-Ramanujan functions.
Research partially supported by grant MDA904-00-1-0015 from the National Security Agency
(Berndt).
Research supported by grant R01-2003-000-11596-0 from the Basic Research Program of the
Korea Science and Engineering Foundation (Y.-S. Choi).
Research partially supported by a grant from the Number Theory Foundation (Yee).
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