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In 2009, M. S. M. Naika et al. obtained Schl a ¨ fli type modular equations of degree 4. Motivated by their work, in the present work, we prove a class of modular equations in Ramanujan's alternative theory of elliptic functions using theory of signature 4. In particular, the composite degrees of modular equations {1, 2, 4}, {1, 3, 9}, {1, 5, 25} and {1, 7, 49}. [ABSTRACT FROM AUTHOR]

Michael Somos used PARI/GP script to generate several Dedekind eta-function identities by using computer. In the present work, we prove two new Dedekind eta-function identities of level 10 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos's Dedekind eta-function identities of level 10 proved by B. R. Srivatsa Kumar and D. Anu Radha. As an application of this, we establish colored partition identities. [ABSTRACT FROM AUTHOR]