Some new generalized modular relations

Abstract

In view of the Rogers-Ramanujan theta function, several researchers have emphasized the subject of integer partitions and their generating functions for years. The work has not dealt with in an inclusive and closed-form by now. Thus, these generating functions are extremely desirable to be developed in a generalized way. In this paper, a novel methodology is proposed to build many new modular relations. By incorporating these new relations, generalized forms of regular partitions in the spirit of Rogers-Ramanujan and G¨ollnitz-Gordon functions with four, six, and nine dissections are established. As an application of these generalized generating functions, an infinite family of new congruences modulo 2 has also been developed.