On Certain Identities for Ratios of Theta-Functions and Some New Modular Equations of Mixed Degree

In this paper, we derive certain identities for ratios of theta-functions. As applications of the identities, we establish certain new modular equations of mixed degree in the theory of signature 3, which are analogous to Ramanujan–Weber type modular equations and Ramanujan–Schläfli type mixed modular equations.