On four generalized basic series and combinatorics

Abstract

In 1996, Agarwal interpreted four generalized basic series in terms of (n-ft)-color partitions. He further extended these results by using weighted lattice paths. In this paper, we give new combinatorial interpretations of these four generalized basic series by the aid of two different combinatorial parameters, viz.t anti-hook differences and associated lattice paths. These new combinatorial interpretations give a new direction to look at the classical partition identities graphically. These results also produce four new classes of infinite 4-way combinatorial identities. Further some interesting basic series identities of Rogers-Ramanujan type are also discussed as the particular cases. These new combinatorial identities reveal the fact that our main results have the potential to yield Rogers-Ramanujan-MacMahon type partition identities linking colored partitions with anti-hook differences and associated lattice paths. © 2019 Utilitas Mathematica Publishing Inc.. All rights reserved.