  The Rogers-Ramanujan recursions are studied from the viewpoint of free
representations over free (generalized) vertex algebras. Specifically, we
construct short exact sequences among the free representations over free
generalized vertex algebras which lift the recursions. They naturally
generalize exact sequences introduced by S. Capparelli et al. We also show that
an analogue of the Rogers-Ramanujan recursions is realized as an exact sequence
among finite-dimensional free representations over a certain finite-dimensional
vertex algebra. As an application of exact sequences, we reveal a relation
between free generalized vertex algebras and $\hat{\mathfrak{sl}}_2$ spaces of
coinvariants $L_{1,0}^{N,\infty}(\mathfrak n)$ introduced by B. Feigin et al.
Moreover, it is shown that generalized $q$-Fibonacci recursions may be realized
as exact sequences obtained by applying several functors to exact sequences
among free representations over free generalized vertex algebras. Finally, we
consider a character decomposition formula of the basic
$\hat{\mathrm{sl}}_2$-module $L_{1,0}$ by M. Bershtein et al. related to the
Urod vertex operator algebras, from the viewpoint of free representations and
exact sequences.
