What is known about doubly exponential series?
I've been exploring functions that have a general form:
$$\sum_{k=0}^\infty{ a^{b^k} } \tag{1}$$
In particular, I'm now checking this equality, which seems to hold:
$$2 \sum_{k=0}^\infty{ \left( \frac{1}{2^{2^k}} - \frac{1}{2^{2^k\cdot3-1}} \right) } = 5/6$$
I'm also in the process of finding more identities/equations, but I don't want to reinvent the wheel.
So I'm wondering, What is known about series of the form (1)? I'm interested in this and anything related to "doubly exponential" series. I'd be extremely interested in any books or papers that anyone knows about.
$\endgroup$ 101 Answer
$\begingroup$The function $$ f(z)=\sum_{k=0}^{\infty}z^{a^k}=z+z^a+z^{a^2}+z^{a^3}+\ldots, $$ where $a$ is a positive integer, is analytic for $|z|<1$, equal to $0$ at $z=0$, and satisfies the functional equation $$ f(z^a)=f(z)-z. $$ For $a=2$, you have the additional fun property that $$ f(z)+f(z^3)+f(z^5)+\ldots=\frac{z}{1-z}. $$
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