A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

Abstract

Our principal second goal is to show that this identity leads to a short proof of Ramanujan’s famous congruence p(11n+6) ≡ 0 (mod 11), where p(n) denotes the number of unrestricted partitions of the positive integer n. Our proof of Theorem 2.2 is short but depends upon some results of Ramanujan from his notebooks [18]. In Section 4, we give a more elementary proof, based upon Ramanujan’s 1ψ1 summation formula, of Ramanujan’s principal result, Lemma 2.1, which is employed in our proof of Theorem 2.2. In Section 5, we give an entirely different and direct proof of Theorem 2.2 based on several elementary identities for theta functions due to Ramanujan. In fact, during our first proof of Theorem 2.2, we establish a special case of a general result on Eisenstein series found in Ramanujan’s lost notebook [19, p. 369]. More precisely, Ramanujan asserts that every member of a certain class of infinite series can be expressed in terms of Ramanujan’s Eisenstein series P , Q, and R (to be defined in Section 6). A less precise version of this claim appears in Ramanujan’s notebooks [18], [2, p. 65, Entry 35(i)], but we prove the better version in Section 6. D. Stanton empirically discovered an analogue of Theorem 2.2, and in Section 7 we give a proof of Stanton’s identity. In Section 8, we prove that (q; q) ∞ is lacunary; in this connection, see a result of J.–P. Serre [20].