  We state and prove product formulae for several generating functions for
sequences $(a_n)_{n\ge0}$ that are defined by the property that $Pa_n+b^2$ is a
square, where $P$ and $b$ are given integers. In particular, we prove
corresponding conjectures of the second author. We show that, by means of the
Jacobi triple product identity, all these generating functions can be reduced
to a linear combination of theta function products. The proof of our formulae
then consists in simplifying these linear combinations of theta products into
single products. We do this in two ways: (1) by using modular function theory,
and (2) by applying the Weierstra\ss addition formula for theta products.
