Some new modular equations in Ramanujan’s alternate theory of signature 3

Abstract

In this paper, we prove some new modular equations in the theory of signature 3 or cubic modular equations by using theta-function identities. Particularly, we prove modular equations of prime degrees 2, 3, 5, 7 and 11. We also prove some modular equations of composite degrees $$\{1, 2, 4\}, \{1, 3, 9\}, \{1, 7, 49\}, \{1, 2, 3, 6\}, \{1, 2, 5, 10\}, \{1, 2, 7, 14\}, \{1, 2, 11, 22\}, \{1, 2, 13, 26\} \{1, 3, 4, 12\}, \{1, 3, 5, 15\}, \hbox { and } \{1, 3, 7, 21\}$$ .