# Ramanujan’s Forty Identities for the Rogers-Ramanujan Functions

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*Bruce C. Berndt; Geumlan Choi; Youn-Seo Choi; Heekyoung Hahn; Boon Pin Yeap; Ae Ja Yee; Hamza Yesilyurt; Jinhee Yi*

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.

#### Table of Contents

# Table of Contents

## Ramanujan's Forty Identities for the Rogers-Ramanujan Functions

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. Definitions and Preliminary Results 512 free
- Chapter 3. The Forty Identities 714
- Chapter 4. The Principal Ideas Behind the Proofs 1522
- Chapter 5. Proofs of 35 of the 40 Entries 2532
- Chapter 6. Asymptotic "Proofs" of Entries 3.28 (second part), 3.29, 3.30, 3.31, and 3.35 8592
- Chapter 7. New Identities for G[sub((q))] and H[sub((q))] and Final Remarks 93100
- Bibliography 95102