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

Abstract

We study a class of well-poised basic hypergeometric series J ˜ k , i ( a ; x ; q ) , interpreting these series as generating functions for overpartitions defined by multiplicity conditions on the number of parts. We also show how to interpret the J ˜ k , i ( a ; 1 ; q ) 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 , q 2 ) , and ( 0 , q ) , where some of the functions J ˜ k , i ( a ; 1 ; q ) become infinite products. The latter case corresponds to Bressoud's family of Rogers–Ramanujan identities for even moduli.