An extension to overpartitions of the Rogers–Ramanujan identities for even moduli

We study a class of well-poised basic hypergeometric series , interpreting these series as generating functions for overpartitions defined by multiplicity conditions on the number of parts. We also show how to interpret the as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases (a,q)→(1/q,q), (1/q,q2), and (0,q), where some of the functions become infinite products. The latter case corresponds to Bressoud's family of Rogers–Ramanujan identities for even moduli.