On Certain Generalizations of Rogers-Ramanujan Type Identities

TL;DR

This paper presents new unilateral and bilateral q-series identities, generalizing classical q-binomial sums and the q-Airy function, revealing symmetries and consequences in basic hypergeometric functions.

Contribution

It introduces novel generalizations of q-series identities, including extensions of the q-binomial sum and q-Airy function, with symmetry properties derived from Heine's transformation.

Findings

New unilateral and bilateral q-series identities

Generalizations of q-binomial sums and q-Airy function

Identified variable-parameter symmetries in hypergeometric functions

Abstract

We state and prove a number of unilateral and bilateral $q$-series identities and explore some of their consequences. Those include certain generalizations of the $q$-binomial sum which also generalize the $q$-Airy function introduced by Ramanujan, as well as certain identities with an interesting variable-parameter symmetry based on limiting cases of Heine's transformation of basic hypergeomteric functions.