Explicit Evaluation Formula for Ramanujan's Singular Moduli and Ramanujan–Selberg Continued Fractions

At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli $\alpha_n$. All those results were proved by Berndt et. al by using Weber–Ramanujan's class invariants. In this paper, we initiate to derive the explicit evaluations formula for $\alpha_{9n}$ and $\alpha_{n/9}$ by involving the class invariant. For this purpose, we establish several new $P-Q$ mixed modular equations involving theta-functions. We apply these modular equations further, deriving a new formula for the explicit evaluation of the Ramanujan–Selberg continued fraction.