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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]

The main objective of this paper is to establish as many as thirty new closed-form evaluations of the generalized hypergeometric function q+1Fq(z) for q = 2; 3. This is achieved by means of separating the generalized hypergeometric function q+1Fq(z) for q = 1; 2; 3 into even and odd components together with the use of several known infinite series involving reciprocal of the non-central binomial coefficients obtained earlier by L. Zhang and W. Ji. [ABSTRACT FROM AUTHOR]

Michael Somos conjectured thousands of theta-functions of various levels with the help of computer and not provided the proof. These theta-functions highly matches Ramanujan's recordings. The objective of our work in this paper is to prove two theta-functions of level 6 conjectured by Somos and to give an alternate proof of the identities proved by B. R. Srivatsa Kumar and G. Sharath. Further, as an application of this, we offer colored partition identities for the same. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function F q + 1 q z for q = 2 , 3 , 4 , 5. This is achieved by means of separating the generalized hypergeometric function F q + 1 q z (q = 2 , 3 , 4 , 5) into even and odd components together with the use of several known infinite series involving reciprocals of the central binomial coefficients obtained earlier by Lehmer. [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]

Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre's arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using (p , k)-parametrization in terms of Borweins' theta functions. [ABSTRACT FROM AUTHOR]

In the present work, we prove few new Dedekind eta-function identities of level 6 discovered bySomos in two different methods. Also during this process, we give an alternate method to Somos's Dedekind eta-function identities of level 6 proved by B. R. Srivatsa Kumar et al. As an application of this, we establish colored partition identities. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate two parameters hk,n and h ' k,n of some P-Q type Theta functions ϕ(q) for any positive real numbers k and n. During this process, we also evaluate Ramanujan's cubic continued fraction. [ABSTRACT FROM AUTHOR]