## Christian Krattenthaler, Mircea Merca and
Cristian-Silviu Radu

# Infinite product formulae for generating functions for
sequences of squares

### (37 pages)

**Abstract.**
We state and prove product formulae for several generating functions
for sequences
(*a*_{n})_{n>=0}
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 the use of modular function theory, and (2) by applying
the Weierstraß addition formula for theta products.

The following versions are available:

Back to Christian Krattenthaler's
home page.