Separable integer partition classes and Slater's list -- I

TL;DR

This paper uses Andrews' separable integer partition classes to interpret and generalize Rogers-Ramanujan type identities from Slater's list, providing new combinatorial insights and alternative expressions.

Contribution

It introduces a systematic SIP framework to interpret and generalize identities, linking series sides to partition theory and deriving new product forms.

Findings

Natural partition interpretations for several identities

Parameterized generalizations of series sides

Alternative expressions reducing to infinite products

Abstract

Slater's list of Rogers-Ramanujan type identities consists of 130 series-product identities whose analytic proofs rely primarily on Bailey pair techniques. Although these identities play an important role in the theory of $q$-series and partitions, combinatorial interpretations for many of them remain unknown, largely because the series sides are difficult to interpret naturally in terms of partitions. In this paper we apply Andrews' theory of separable integer partition (SIP) classes to several identities from Slater's list. By constructing suitable SIP classes, we obtain natural partition-theoretic interpretations and parameterized generalizations of their series sides. We then apply various $q$-hypergeometric transformations to these generalized series to derive alternative expressions, which in certain cases reduce to infinite products. These results illustrate how the SIP framework provides a systematic approach to understanding Rogers-Ramanujan type identities and offer new combinatorial insights into identities appearing in Slater's list.