New Modular Equations of Composite Degrees and Partition Identities

In a recent study, Kim established a general identity which implies a generalization of the modular equations of degrees 3, 5, 11 and 23, and derived some identities for partitions. In this paper we provide proofs for some new modular equations of composite degrees and degree of 7 by methods of elementary algebra and Kim’s generalization of theta-function identities. In addition, we derive many partition identities, which are proved depending upon these modular equations and reciprocation.