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I want to find the use of $q=e^{2\pi i z}$ in modular forms and combinatorics. The question here maybe a little not specific, and I'll explain it.

  • I know that this is a exponential map，and map upper half plane $\mathcal{H}$ into a punctured disk. Besides, it is $\mathbb{Z}$-periodic holomorphic map. Thus, it makes modular forms which are complex functions fall into the framework of $q$-series.

However, I care about the congruent property of $q$-serises and I want to know whether there is algebraiclly convenient to change the $q$ into $e^{2\pi i z}$. To be specific, I just want more example of the use the transformation of $q=e^{2\pi i z}$ in combinatorics. Any help will be appreciated.:)

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    I'm not sure I understand your question. What does "I want to know whether there is algebraiclly convenient to change the $q$ into $e^{2\pi i z}$" mean? – 2017-01-06
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    I may not explain my question well... Sometimes we change $e^{2\pi i z}$ into $q$, just because it is convenient to calculate. I want to know that when dealing with $q$-series, if I change $q$ back to $e^{2\pi i z}$, is there any unexpected benifits？ Or, I just want more examples for this substitution. Thanks for your commend. :) – 2017-01-06

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