The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators

Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for $\widehat{\text{goth}sl(2)}$ satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon’s generalization of the Rogers–Ramanujan identities and with Andrews’ related identities. The present work generalizes the authors’ previous work on intertwining operators and the Rogers–Ramanujan recursion.