We define two quotients of theta-functions depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function and the RamanujanWeber class invariants. Explicit formulas for determining values of the theta-function is derived, and several examples will be given and using them, we give some complete explicit results for the complete elliptic integral of the first kind and the Gaussian hypergeometric function. Also several new modular equations for the theta-function are derived.

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We define two quotients of theta-function ψ depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function, theta-function ϕ, and the Ramanujan-Weber class invariants. Explicit formulas for determining values of the thetafunction ψ are derived, and several examples will be given. In addition, we give some applications of these parameters for the famous Rogers-Ramanujan continued fraction R(q), Ramanujan's cubic continued fraction G(q), and the modular j -invariant.

Abstract. We first derive several modular equations of degree 5 and present their concise proofs based on algebraic computations. We then establish explicit relations and formulas for some parameterizations for the theta functions ϕ and ψ by using the derived modular equations. In addition, we find specific values of the parameterizations and evaluate some numerical values of the Rogers-Ramanujan continued fraction.

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