Some identities for the partition function

Abstract

In his unpublished manuscript on the partition and tau functions, Ramanujan obtained several striking congruences for the partition function p(n), the number of unrestricted partitions of n. The most notable of them are p(5n+4)≡0(mod5) and p(7n+5)≡0(mod7) which holds for all positive integers n. More surprisingly, Ramanujan obtained certain identities between q-series from which the above congruences follow as consequences. In this paper, we adopt Ramanujan's approach and prove an identity which witnesses another famous Ramanujan congruence, namely, p(11n+6)≡0(mod11) and also establish some new identities for the generating functions for p(17n+5),p(19n+7) and p(23n+1). We also find explicit evaluations for Fp(q) in the cases p=17,19,23 where Fp is the function appearing in Ramanujan's circular summation formula.