Infinite product formulae for generating functions for sequences of squares

We state and prove product formulae for several generating functions for sequences $(a_n{)}_{n\ge 0}$ that are defined by the property that $P{a}_n+b^2$ 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.