What is known about the transformation of a power series in which $z^n$ is replaced with $z^{n^2}$? Say we have the function $$G(z) = \sum_{n \geq 0} g_n z^n.$$  
Is there a name for the transform T defined so that $$(T(G))(z) = \sum_{n \geq 0} g_n z^{n^2}?$$
Is there anything known about this kind of transformation?

If you know a formula for the ordinary generating function of the sequence and its $j^{th}$ derivatives, which must exist for all $j \geq 0$, then this article (2017) provides you with an integral representation of the transformed series in question. In particular, if $G(z)$ is the ordinary generating function of the sequence $\{g_n\}_{n \geq 0}$ and $q \in \mathbb{C}$ is such that $0 < |q| < 1$, then we have proved in the article that 
$$\sum_{n \geq 0} g_n q^{n^2} z^n = \frac{1}{\sqrt{2\pi}} \int_0^{\infty} 
     \left[\sum_{b = \pm 1} G\left(e^{bt \sqrt{2\log(q)} z}\right)\right] e^{-t^2 / 2} dt. $$ The article terms this general procedure for modifying the original sequence generating function a square series transformation integral, but more generally, some of the most interesting applications of this method include new integral representations for theta functions and classical identities such as the series expansion for Jacobi's triple product. 
