Is $$\phi(x):=\sum_{n=0}^\infty x^{2^n}$$ a known special function ?
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4It is an example of a lacunary function. – Jair Taylor Jul 15 '19 at 06:15
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@JairTaylor: yes, but does it have a name and known properties ? – Jul 15 '19 at 06:39
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I'm afraid I don't know anything else other than that reference. – Jair Taylor Jul 15 '19 at 15:13
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To get a feeling of the behaviour for $x$ close to 1 we replace the sum by an integral giving $\phi(x)=-\frac{\text{Ei}(\log (x))}{\log (2)}$, the asymptotic for $x\to 1$ f which is $\frac{-\log (1-x)-\gamma }{\log (2)} + O(1-x)$ – Dr. Wolfgang Hintze Oct 01 '19 at 10:53
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It is interesting to consider the integral of the function. It obviously exists, and it has a closed expression: $\int_0^1 \phi(x) ,dx = \sum _{n=0}^{\infty } \frac{1}{2^n+1}=-1+\frac{1}{\log (2)}\psi _{\frac{1}{2}}^{(0)}\left(-\frac{i \pi }{\log (2)}\right) \simeq 1.2645 $ – Dr. Wolfgang Hintze Oct 01 '19 at 17:08
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Linking to :[|||||]:. – metamorphy Aug 17 '22 at 06:03
2 Answers
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Not an answer but just a note on $x\to 1$ (following [1] and [2] closely).
Basically, $\phi(x)=-\log_2(1-x)+\mathcal{O}(1)$ with the "$\mathcal{O}(1)$" oscillating: $$\psi(x)=\phi(x)+\frac{\ln(-\ln x)}{\ln 2}+\sum_{n=1}^{\infty}\frac{(\ln x)^n}{n!(2^n-1)}$$ satisfies $\psi(x)=\psi(x^2)$. The "Mellin transform approach" gives, in our case, $$\psi\big(e^{-2^{-x}}\big)=\frac{1}{2}-\frac{\gamma}{\ln 2}+\frac{1}{\ln 2}\sum_{n\in\mathbb{Z}\setminus\{0\}}\Gamma\Big(\frac{2n\pi i}{\ln 2}\Big)e^{2nx\pi i}.$$
metamorphy
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There are a few papers by Ahmed Sebbar where he expresses this function in terms of more well known modular forms. (though the expressions are fairly ungainly). There is a relationship with paperfolding and with automatic sequences
On Two Lacunary Series and Modular Curves
Paperfolding and modular functions
This paper came out a few days ago:
Automatic sequences defined by Theta functions and some infinite products by Shuo Li
The series $\sum_{n=1}^{\infty} \frac{1}{2^{2^{n}}}$ is in some contexts known as The Kempner number, and it is known to be transcendental:
In The Many Faces of the Kempner Number, Boris Adamczewski relates a few proofs of its transcendentality.
graveolensa
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