Congruences for $\ell$-regular partitions and bipartitions

Abstract

Define $F\left(q\right):={\sum }_{n=-\infty }^{\infty }{\left(-1\right)}^{\delta n}\left(an+b\right)q^{\left(c{n}^2+dn\right)∕2}$, 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 $F\left(q\right)$. As an application we deduce several infinite families of congruences for $\ell$-regular partitions and $\ell$-regular bipartitions. In addition, we give a new proof of Ramanujan’s congruence for the unrestricted partition function modulo $5$.