Bulletin of the
Korean Mathematical Society BKMS

In this paper we introduce and study the lattice paths for which the horizontal step is allowed at height $h\geq0$, $h\in \mathbb{Z}$. By doing so these paths generalize the heavily studied weighted lattice paths that consist of horizontal steps allowed at height zero only. Six $q$--series identities of Rogers--Ramanujan type are studied combinatorially using these generalized lattice paths. The results are further extended by using $(n+t)$--color overpartitions. Finally, we will establish that there are certain equinumerous families of $(n+t)$--color overpartitions and the generalized lattice paths.

Keywords: \(q\)--series, generalized lattice paths, \((n+t)\)--color overpartitions, combinatorial identities

MSC numbers: Primary 05A15, 05A17, 05A19, 11P81