CHAPTER 3 The Forty Identities ENTRY 3.1. (3.1) GU(q)H(q) ~ q2G(q)HU(q) = 1 + llqG\q)H\q). Entry 3.1 is one of two identities stated by Ramanujan without proof in [26], [27, p. 231]. As related in the Introduction, Ramanujan [26] claims that, "Each of these formulae is the simplest of a large class." Ramanujan's remark is interesting, because Entry 3.1 is the only identity among the forty in which powers of G{q) or H(q) appear. It would seem from Ramanujan's remark that he had further identities involving powers of G(q) or H(q), but no further identities of this sort are known. The first published proof of (3.1) is by H. B. C. Darling [16] in 1921. A second proof by Rogers [32] appeared in the same year. One year later, L.J. Mordell [24] found another proof. By (2.21), the identity (3.1) is equivalent to a famous identity for the Rogers- Ramanujan continued fraction (2.20), namely, (3.2) ^ - 11 - R3(q) ~ f ( _ ( ? ) R5(q) w qf6(-Q5Y This equality was found by Watson in Ramanujan's notebooks [28] and proved by him [33] in order to establish claims about the Rogers-Ramanujan continued frac- tion communicated by Ramanujan in his first two letters to Hardy [33], A different proof of (3.2) can be found in Berndt's book [5, pp. 265-267]. The identity (3.2) can also be found in an unpublished manuscript of Ramanujan first appearing in handwritten form with his lost notebook [29, pp. 135-177, 238-243]. An annotated account of Ramanujan's manuscript with considerable commentary and numerous references has been prepared by Berndt and K. Ono [7]. ENTRY 3.2. 2/ ^ v(q) (3.3) G{q)G{q*) + qH(q)H(q*)=X2(q)- f{ _q2y Entry 3.2 was first proved in print by Rogers [32] Watson [34] also found a proof. In fact, G. E. Andrews [1, p. 27] has shown that (3.3) follows from a very general identity in three variables found in Ramanujan's lost notebook. ENTRY 3.3. (3.4) G(q)G(q4)-qH(q)H(q^- ^ f(-Q2Y Watson [34] gave a proof of (3.4). ENTRY 3.4. (3.5) G(q11)H(q)-q2G(q)H(q11) = l. 7
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