  We consider new series expansions for variants of the so-termed ordinary
geometric square series generating functions originally defined in the recent
article titled "Square Series Generating Function Transformations" (arXiv:
1609.02803). Whereas the original square series transformations article adapts
known generating function transformations to construct integral representations
for these square series functions enumerating the square powers of $q^{n^2}$
for some fixed non-zero $q$ with $|q| < 1$, we study the expansions of these
special series through power series generated by Jacobi-type continued
fractions, or J-fractions. We prove new exact expansions of the $h^{th}$
convergents to these continued fraction series and show that the limiting case
of these convergent generating functions exists.
  We also prove new infinite $q$-series representations of special square
series expansions involving square-power terms of the series parameter $q$, the
$q$-Pochhammer symbol, and double sums over the $q$-binomial coefficients.
Applications of the new results we prove within the article include new
$q$-series representations for the ordinary generating functions of the special
sequences, $r_p(n)$, and $\sigma_1(n)$, as well as parallels to the examples of
the new integral representations for theta functions, series expansions of
infinite products and partition function generating functions, and related
unilateral special function series cited in the first square series
transformations article.
