Abstract
We investigate class of well-poised basic hypergeometric series $\tilde{J}_{k,i}(a;x;q)$, interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the $\tilde{J}_{k,i}(a;1;q)$ 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 $(a,q) \to (1/q,q)$, $(1/q,q^2)$, and $(0,q)$, where some of the functions $\tilde{J}_{k,i}(a;x;q)$ become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.
Highlights
Over the years, a great number of combinatorial identities [1, 2, 3, 4, 8, 10, 19, 23, 25] have been extracted fromAndrews’ functions [7, Ch. 7] Â Ü Õμ, which are defined byÂ Ü Õμ À ÜÕ Õμ · ÜÕÀ 1⁄2 ́ ÜÕ Õμ (1.1) where À Ü Õμ Ò1⁄4μÒÕ Ò3⁄4·Ò ÒÜ Ò1⁄2 Ü Õ3⁄4Ò μ 1⁄2 μÒ ́ÕμÒÜÕÒμ1⁄2 ÜÕÒ·1⁄2μ1⁄2 (1.2)Here we have employed the usual basic hypergeometric series notation [21]
It was shown that the  1⁄2 Õμ can be interpreted as generating functions for overpartitions with bounded successive ranks, for overpartitions with a specified Durfee dissection, and for certain restricted lattice paths
In this paper we study a similar class of functions, which we call Â Ü Õμ and define by μÒÕ Ò3⁄4 ́Ò3⁄4μ·Ò ÒÜ 1⁄2μÒ1⁄2 Ü Õ3⁄4Ò μ ÜÕ3⁄4 Õ3⁄4μÒÜÕÒμ1⁄2
Summary
A great number of combinatorial identities [1, 2, 3, 4, 8, 10, 19, 23, 25] have been extracted from. Ñ define the function Ñ Òμ to be the number of overpartitions of Ò with Ñ parts, of which are overlined, such thatμ 1⁄2 ́ μ· andμ if is saturated at , that is, if the maximum in 1⁄2μisμachieve1⁄2d, , ́. 3⁄4, let ¿ ́Òμ denote the number of overpartitions whose non-overlined parts are not. In the second half of the paper, we discuss three more combinatorial interpretations of the  1⁄2 Õμ: one involving the theory of successive ranks for overpartitions as developed in [19], one involving a two-parameter generalization to overpartitions of Garvan’s -conjugation for partitions [20], and one involving a generalization of some lattice paths of Bressoud and Burge [14, 15, 16].
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