Congruences for $\ell$-regular partitions and bipartitions

Abstract

Define F ( q ) : = ∑ n = − ∞ ∞ ( − 1 ) δ n ( a n + b ) q ( c n 2 + d n ) ∕ 2 , which includes Ramanujan’s theta function as a special case. We establish a dissection identity for this function, and use it to derive congruence properties for the coefficients of F ( q ) . As an application we deduce several infinite families of congruences for ℓ -regular partitions and ℓ -regular bipartitions. In addition, we give a new proof of Ramanujan’s congruence for the unrestricted partition function modulo 5 .