Transformation of Products of υ-Functions

We shall adopt the following well-known notations for υ-functions:

q = e^πiτ, Imτ > o ;

υ₂ (x|τ) = Σ q^{(n+1/2)2(2n+2)xπi},

υ (x|τ) = Σ qⁿ² e^2nxπi,

υ (x|τ) = Σ (-I)ⁿqⁿ² e^2nπi.

Here, as well as in the sequel, Σ denotes summation over all integers from -∞ to +∞.