Series and polynomial representations for weighted Rogers-Ramanujan partitions and products modulo 6

Abstract

Infinite series representations are now obtained for certain weighted Rogers-Ramanujan partitions which we recently showed are related to partitions into $\mathrm{parts} \not\equiv 0$, $\pm i$ ($\mathrm{mod}\ 6$), for $i = 1, 2, 3$. We also show that our series can be transformed to the series previously obtained by Bressoud which connect the partitions into $\mathrm{parts} \not\equiv 0 \pm i$ ($\mathrm{mod}\ 6$) with partitions satisfying certain bounds on their successive ranks. Finally, we obtain finite versions of our series representations, namely, polynomial identities which tend to the infinite series identities when certain parameters tend to infinity.