A new parameter for Ramanujan's theta-functions and explicit values

We define a new parameter Ak,n involving Ramanujan's theta-functions ϕ(q) and ψ(q) for any positive real numbers k and n and study its several properties. We also prove some general theorems for the explicit evaluations of the parameter Ak,n and find many explicit values. Finally, we establish an explicit formula for values of ψ(e−2nπ) for any positive real number n in terms of Ak,n and give examples.