# Continued Fraction Proofs of $m$-versions of Some Identities of Rogers-Ramanujan-Slater Type

## TL;DR

This paper develops new transformations for hypergeometric series using $q$-continued fractions, leading to novel and existing $m$-versions of Rogers-Ramanujan-type identities through parameter specialization and classical transformations.

## Contribution

It introduces general transformations for hypergeometric series based on $q$-continued fractions, enabling derivation of new and known $m$-versions of Rogers-Ramanujan identities.

## Key findings

- Derived new $m$-versions of Rogers-Ramanujan identities
- Reproduced known identities using different methods
- Extended identities via classical transformations

## Abstract

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive \emph{$m$-versions} of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such $m$-versions of Rogers-Ramanujan-type identities.