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At the end of his brief communication [7], [8, 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.: H(q){G(q)}11 − q2G(q){H(q)}11 = 1 + 11q{G(q)H(q)}6. (1.3) Another noteworthy formula is H(q)G(q11)− q2G(q)H(q11) = 1. (1.4) Each of these formulae is the simplest of a large class.” In a manuscript of Ramanujan, published with his Lost Notebook [9], there are forty identities involving the Rogers–Ramanujan functions. In 1921, L. J. Rogers [10] established ten of the identities including (1.3) and (1.4). In his paper [12] proving eight of the identities, but with two of them from the list that Rogers proved, Watson confides, “Among the formulae contained in the manuscripts left by Ramanujan is the set of about forty which involve functions of the types G(q) and H(q); the beauty of

Septic analogues of the Rogers–Ramanujan functions