  We construct new integral representations for transformations of the ordinary
generating function for a sequence, $\langle f_n \rangle$, into the form of a
generating function that enumerates the corresponding "square series"
generating function for the sequence, $\langle q^{n^2} f_n \rangle$, at an
initially fixed non-zero $q \in \mathbb{C}$. The new results proved in the
article are given by integral-based transformations of ordinary generating
function series expanded in terms of the Stirling numbers of the second kind.
We then employ known integral representations for the gamma and double
factorial functions in the construction of these square series transformation
integrals. The results proved in the article lead to new applications and
integral representations for special function series, sequence generating
functions, and other related applications. A summary Mathematica notebook
providing derivations of key results and applications to specific series is
provided online as a supplemental reference to readers.
