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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]

S. Ramanujan documented several modular equations of degrees in his notebooks. These identities are used to evaluate Weber's class in variants, continued fractions and many more. In the present work, we establish modular equations of composite degrees using the known identities. [ABSTRACT FROM AUTHOR]

M. Somos discovered around 6200 theta function identities of various levels by using PARI/GP script. He has not offered any proof for these identities. In this paper, we prove Somos's theta-function identities of level 6 by using modular equation of degree 3. [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]

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]