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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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    Well, they become lacunary most of the time, and z usually becomes restricted to be within the unit disk after such a change.2010-11-12
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    The original series does not really bear a strong relation to the new series. I don't think this is a useful definition.2010-11-12
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    @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: http://en.wikipedia.org/wiki/Lacunary_function2010-11-12
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    @Eric: it really doesn't. For example, the "transform" of the innocent rational function 1/(1-z) is a much more complicated beast: http://en.wikipedia.org/wiki/Theta_function2010-11-13
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    @Qiaochu Yuan: There might be good applications in Hilbert space theory or on $\ell^p(\mathbb{Z}_+)$, where the map is linear and norm preserving (considering $z^n$ as a 'base').2010-11-13
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    @AD.: it doesn't preserve any multiplicative structure.2010-11-13
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    @Qiaochu Yuan: Well, on $\ell^1$ it is multiplicative. Hence, we do at least have that $r(f)=r(G(f))$ where $r(f)$ is the spectral radius of $f$.2010-11-13
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    @AD.: right, but that's true of any transformation where one replaces z^n with z^{f(n)} for some f. The vast majority of these transformations do not admit interesting analytic descriptions and are - I will say it again - not useful. It will take a significant application to convince me otherwise.2010-11-13
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    @Qiaochu Yuan: Sure, you may be right. However, to me it is not obvious that there is no application. :)2010-11-13
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    What I had in mind were combinatorial applications -- in particular replacing $z^n$ with $z^\binom{n}{p}$.2010-11-19

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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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    links seems to be broken now2018-11-07
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    Fixed the link to the arxiv version. Thanks.2018-11-07
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    Inside $G$ it should be $e^{bt \sqrt{2\log(q)}}z$, right?2018-11-26
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    Yes, fixing it now.2018-11-27