Аннотация:
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.
Ключевые слова:modular equations, singular moduli, continued fraction.