Some identities for the partition function - Goswami, A; Jha, A K; Singh, A K (Academic Press Inc., 2021)
In his unpublished manuscript on the partition and tau functions, Ramanujan obtained several striking congruences for the partition function p(n), the number of unrestricted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.
This work was done when the first author was a postdoc at Research Institute for Symbolic Computation (RISC), Austria. He was supported by grant SFB F50-06 of the Austrian Science Fund (FWF). The authors thank the anonymous referee for the comments and feedback. The first author thanks Ralf Hemmecke for explaining how the function GroebnerBasis() in Mathematica works. The authors thank Madeline Locus Dawsey, Atul Dixit, Frank Garvan, Ben Kane, Robert Osburn and Peter Paule for their feedback. Our computation have been done in SageMath [34] and Mathematica [
Keywords: Modular forms; Partition function; Ramanujan's circular summation; Ramanujan's congruences; Witness identity