On combinatorial extensions of Rogers-Ramanujan type identities

Abstract

In the present paper we use anti-hook differences of Agarwal and Andrews as an elementary tool to provide new partition theoretic meanings to two generalized basic series in terms of ordinary partitions satisfying certain anti-hook difference conditions. Five particular cases are also discussed. These particular cases yield new partition theoretic versions of G\{o}llnitz-Gordon identities and G\{o}llnitz identity. Five $q$-identities of Rogers and three $q$-identities of Slater are further explored. These results extend the work of Goyal and Agarwal, Agarwal and Rana and Sareen and Rana.