Infinite product formulae for generating functions for sequences of squares

TL;DR

This paper derives infinite product formulas for generating functions of sequences where each term's property relates to being a square, using theta functions and modular identities, confirming several conjectures.

Contribution

The paper introduces new infinite product formulas for specific generating functions and proves several conjectures using theta function identities and modular function theory.

Findings

Derived explicit product formulas for generating functions.

Reduced generating functions to theta function products.

Validated conjectures through modular and addition formulas.

Abstract

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.