Abstract
We define two classes of multiple basic hypergeometric series $V_{k,t}(a,q)$ and $W_{k,t}(a,q)$ which generalize multiple series studied by Agarwal, Andrews, and Bressoud. We show how to interpret these series as generating functions for special restricted lattice paths and for $n$-color overpartitions with weighted difference conditions. We also point out that some specializations of our series can be written as infinite products, which leads to combinatorial identities linking $n$-color overpartitions with ordinary partitions or overpartitions. Nous définissons deux classes de séries hypergéométriques basiques multiples $V_{k,t}(a,q)$ et $W_{k,t}(a,q)$ qui généralisent des séries multiples étudiées par Agarwal, Andrews et Bressoud. Nous montrons comment interpréter ces séries comme les fonctions génératrices de chemins avec certaines restrictions et de surpartitions $n$-colorées vérifiant des conditions de différences pondérées. Nous remarquons aussi que certaines spécialisations de nos séries peuvent s'écrire comme des produits infinis, ce qui conduit à des identités combinatoires reliant les surpartitions $n$-colorées aux partitions ou surpartitions ordinaires.
Highlights
Many multiple series linked to partitions and related objects have been discovered, such as Andrews’ generalization of the Rogers-Ramanujan identities (8) or an infinite family which was studied a few years ago in (6)
Another combinatorial interpretation of these multiple series uses a family of lattice paths that appeared in (14) to interpret a generalization of the Andrews-Gordon identities to overpartitions: n-color overpartitions, lattice paths, and multiple basic hypergeometric series
Such a change would increase m by the number of South steps between our two peaks, which is the number of NES peaks between (n, j) included and (m, i) excluded, or, in the corresponding n-color overpartition, the number of overlined parts whose size lies in the interval [n, m)
Summary
Many multiple series linked to partitions and related objects have been discovered, such as Andrews’ generalization of the Rogers-Ramanujan identities (8) or an infinite family which was studied a few years ago in (6). The special case a → 0 of these series was studied analytically in (3); it was interpreted combinatorially in (2) using n-color partitions and in (4) using lattice paths. Theorem 1.2 Wk,t(a, q) is the generating function for (n + t)-color overpartitions counted by Vk,t(a, q) such that no part of the form xx+t is overlined Another combinatorial interpretation of these multiple series uses a family of lattice paths that appeared in (14) to interpret a generalization of the Andrews-Gordon identities to overpartitions: Theorem 1.3 Vk,t(a, q) is the generating function for lattice paths which start at (0, t) and have no valley above height k − 3 (or no valley at all if k = 2) where the exponent of q counts the major index and that of a counts the number of South steps.
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