Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

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 functions, 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 devised 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, the authors 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, the authors present two or three proofs. For the five identities that they are unable to prove, they provide non-rigorous verifications based on an asymptotic analysis of the associated Rogers-Ramanujan functions. This method, which is related to the 5-dissection of the generating function for cranks found in Ramanujan'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.