Some New Congruences and Partition-Theoretic Interpretations for the Coefficients of Some Rogers-Ramanujan Type Identities

TL;DR

This paper provides new partition-theoretic interpretations for Rogers-Ramanujan type identities and establishes infinite families of congruences modulo powers of 2, enhancing understanding of these classical q-series identities.

Contribution

It introduces novel partition interpretations for Rogers-Ramanujan identities and proves new congruences, expanding the combinatorial and number-theoretic understanding of these identities.

Findings

Partition-theoretic interpretations using overpartitions and color partitions.

Infinite families of congruences modulo powers of 2.

Deeper insight into Rogers-Ramanujan type identities.

Abstract

Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we give partition-theoretic interpretations of some of the Rogers-Ramanujan type identities using overpartition and colour partition of positive integers, and prove infinite families of congruences modulo powers of 2.