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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?

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    $\begingroup$ Well, they become lacunary most of the time, and z usually becomes restricted to be within the unit disk after such a change. $\endgroup$ Commented Nov 12, 2010 at 23:25
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    $\begingroup$ The original series does not really bear a strong relation to the new series. I don't think this is a useful definition. $\endgroup$ Commented Nov 12, 2010 at 23:38
  • $\begingroup$ @J.M. I don't think it actually becomes lacunary in this case. I think that it pretty much has the same properties as the original series: en.wikipedia.org/wiki/Lacunary_function $\endgroup$ Commented Nov 12, 2010 at 23:49
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    $\begingroup$ @Eric: it really doesn't. For example, the "transform" of the innocent rational function 1/(1-z) is a much more complicated beast: en.wikipedia.org/wiki/Theta_function $\endgroup$ Commented Nov 13, 2010 at 0:15
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    $\begingroup$ @Qiaochu Yuan: Sure, you may be right. However, to me it is not obvious that there is no application. :) $\endgroup$ Commented Nov 13, 2010 at 11:55

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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.

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  • $\begingroup$ links seems to be broken now $\endgroup$
    – Andrew
    Commented Nov 7, 2018 at 0:50
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    $\begingroup$ Fixed the link to the arxiv version. Thanks. $\endgroup$
    – mds
    Commented Nov 7, 2018 at 1:02
  • $\begingroup$ Inside $G$ it should be $e^{bt \sqrt{2\log(q)}}z$, right? $\endgroup$
    – Andrew
    Commented Nov 26, 2018 at 17:12
  • $\begingroup$ Yes, fixing it now. $\endgroup$
    – mds
    Commented Nov 27, 2018 at 0:04

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