Rogers-Ramanujan type generalized reciprocal identities and Eisenstein series

In a recent paper Huber and Schultz have given relations between eta quotients and Rogers-Ramanujan type generalized reciprocal identities. Inspired by that work, we study the relationship between the Eisenstein series and the level p generalized reciprocal identities, where p is a prime that is congruent to 1 modulo 4, by employing techniques from modular forms. In the aforementioned paper, Huber and Schultz additionally point to a relationship between these reciprocals and the class number of the field Q(root p). Our methods allow us to give this relationship explicitly by obtaining a formula for the class number of the field Q(root p) which involves Fourier coefficients of these reciprocals.