In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21. These equations can be regarded as modular identities in the alternate theory of signature 3. As a consequence, several values of quotients of theta--function are evaluated.
Keywords and Phrases
Modular equations, Theta-functions.
A.M.S. subject classiﬁcation
33D15, 33E20, 11F20, 11F27.