# $q$-Difference Equations and Identities of the Rogers--Ramanujan--Bailey Type

## TL;DR

This paper introduces $q$-difference equations linked to the standard multiparameter Bailey pair, enabling the derivation of entire families of Rogers-Ramanujan type identities, expanding the understanding of these identities in combinatorics.

## Contribution

It establishes $q$-difference equations for the SMPBP and uses them to generate comprehensive families of Rogers-Ramanujan type identities, revealing new connections.

## Key findings

- Derived $q$-difference equations for SMPBP
- Generated complete families of Rogers-Ramanujan identities
- Connected classical and new identities through these equations

## Abstract

In a recent paper, I defined the "standard multiparameter Bailey pair" (SMPBP) and demonstrated that all of the classical Bailey pairs considered by W.N. Bailey in his famous paper (\textit{Proc. London Math. Soc. (2)}, \textbf{50} (1948), 1--10) arose as special cases of the SMPBP. Additionally, I was able to find a number of new Rogers-Ramanujan type identities. From a given Bailey pair, normally only one or two Rogers-Ramanujan type identities follow immediately. In this present work, I present the set of $q$-difference equations associated with the SMPBP, and use these $q$-difference equations to deduce the complete families of Rogers-Ramanujan type identities.