A generalisation of two partition theorems of Andrews

Abstract

In 1968 and 1969, Andrews proved two partition theorems of the Rogers-Ramanujan type which generalise Schur’s celebrated partition identity (1926). Andrews’ two generalisations of Schur’s theorem went on to become two of the most influential results in the theory of partitions, finding applications in combinatorics, representation theory and quantum algebra. In this paper we generalise both of Andrews’ theorems to overpartitions. The proofs use a new technique which consists in going back and forth from $q$-difference equations on generating functions to recurrence equations on their coefficients.

Keywords

• integer partitions
• overpartitions
• $q$-difference equations
• recurrences
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]