Generalized Rogers Ramanujan Identities from AGT Correspondence

AGT correspondence and its generalizations attracted a great deal of attention recently. In particular it was suggested that $U(r)$ instantons on $R^4\mathrm{/}{Z}_p$ describe the conformal blocks of the coset ${\cal A}(r,p)=U(1)\times sl(p)_r\times {sl®_p\times sl®n\over sl®{n+p}}$,where$n$ is a parameter. Our purpose here is to describe Generalized Rogers Ramanujan (GRR) identities for these cosets, which expresses the characters as certain $q$ series. We propose that such identities exist for the coset $\mathcal{A}(r,p)$ for all positive integers $n$ and all $r$ and $p$. We treat here the case of $n=1$ and $r=2$, finding GRR identities for all the characters.