Certain quotient of eta-function identities.

Abstract

On page 212 in his lost notebook, Ramanujan defined a parameter $\lambda_n$ by a certain quotient of Dedekind eta-functions at the argument $q=exp(-\pi \sqrt{n/3})$. He then recorded a table of several values of $\lambda_n:=\lambda_{n,\,\, 3}$. All these have been established by B. C. Berndt, H. H. Chan, S.-Y. Kang and L.-C. Zhang \cite{BCB4}. In this paper following Ramanujan we defined a parameter $\lambda_{n, p}$ at the argument $q=exp(-\pi \sqrt{n/p})$. We establish several interesting and new explicit evaluations for $\lambda_{n,\,\, p}$ using Ramanujan-Weber class invariant, modular equations and mixed-modular equations