The explicit formulas and evaluations of Ramanujan’s theta-function ψ

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. © 2005 Elsevier Inc. All rights reserved