Basis partition polynomials, overpartitions and the Rogers–Ramanujan identities

In this paper, a common generalization of the Rogers–Ramanujan series and the generating function for basis partitions is studied. This leads naturally to a sequence of polynomials, called BsP-polynomials. In turn, the BsP-polynomials provide simultaneously a proof of the Rogers–Ramanujan identities and a new, more rapidly converging series expansion for the basis partition generating function. Finally the basis partitions are identified with a natural set of overpartitions.