New Theta-Function Identities of Level $6$ in the Spirit of Ramanujan

Michael Somos discovered several theta-function identities of various levels by computer and offered no proof for them. These identities highly resemble some of Ramanujan's identities. The main focus of this paper is to prove some of these theta-function identities, in particular those of level $6$ that have been discovered using computational searches. Some of the the Somos identities that we are discussing in this paper cannot be expressed in the form of $P--Q$ type. Furthermore, we establish certain colored partition identities for them.