CHAPTER 1

Introduction

The Rogers-Ramanujan functions in the title are defined for \q\ 1 by

where here and in the sequel we use the customary notation (a; q)o := 1,

n - l

fc=0

(a; 9)00 := lim (a;g)n, |g| 1.

n—oo

These functions satisfy the famous Rogers-Ramanujan identities [31], [25], [27,

pp. 214-215]

(1.2)

G

(q) = -—^—-—- and H(q) * ta

?5)oo(g4; g5)oo 0?2; 45)oo(?3; g5)oo'

At the end of his brief communication [26], [27, p. 231] announcing his proofs of

the Rogers-Ramanujan identities (1.2), Ramanujan remarks, "I have now found an

algebraic relation between G(q) and H(q), viz.:

(1.3) H(q) {G(q)}U - q2G(q) {H(q)}U = 1 + llq {G(q)H(q)}6 .

Another noteworthy formula is

(1.4) H(q)G(q11)-q2G(q)H(qll) = l.

Each of these formulae is the simplest of a large class." Ramanujan did not indicate

how he had proved these two identities, which, as we shall see below, are two from

a list of forty identities involving G(q) and H(q) that Ramanujan had compiled.

In his paper [32] establishing ten of the identities, Rogers remarks, "these

[identities] were communicated privately to me in February 1919 ..." Rogers did

not indicate if further identities were included in Ramanujan's communication to

him. We remark that Ramanujan returned to India on February 27, 1919, and

that the short paper [26] was recorded in the minutes of the London Mathematical

Society on March 13, 1919. Thus, both the paper to the London Mathematical

Society and the letter to Rogers were evidently sent only days prior to Ramanujan's

departure.

In 1933, Watson [34] proved eight of the identities, but with two of them

from the group that Rogers had proved. Watson confides, "Among the formulae

contained in the manuscripts left by Ramanujan is a set of about forty which involve

functions of the types G(q) and H(q)\ the beauty of these formulae seems to me

to be comparable with that of the Rogers-Ramanujan identities. So far as I know,

nobody else has discovered any formulae which approach them even remotely; if

l