One-Parameter Generalizations of Rogers-Ramanujan Type Identities

TL;DR

This paper introduces new one-parameter generalizations of Rogers-Ramanujan type identities using recursion relations and determinant methods, expanding the scope of classical partition identities.

Contribution

It presents novel one-parameter generalizations of Rogers-Ramanujan identities, including extensions of Göllnitz-Gordon and Rogers-Selberg identities, using recursion and determinant techniques.

Findings

Derived many new one-parameter identities

Extended classical partition identities

Connected identities through recursion and determinants

Abstract

Resorting to the recursions satisfied by the polynomials which converge to the right hand sides of the Rogers-Ramanujan type identities given by Sills and a determinant method presented in a paper by Ismail-Prodinger-Stanton, we obtain many new one-parameter generalizations of the Rogers-Ramanujan type identities, such as a generalization of the analytic versions of the first and second G\"{o}llnitz-Gordon partition identities, and generalizations of the first, second, and third Rogers-Selberg identities.